Calculus Examples

$lim_{x \to 0} \frac{x^{2} e^{x}}{\tan^{2} (x)}$

Step 1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{lim_{x \to 0} x^{2} e^{x}}{lim_{x \to 0} \tan^{2} (x)}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{lim_{x \to 0} x^{2} \cdot lim_{x \to 0} e^{x}}{lim_{x \to 0} \tan^{2} (x)}$

Step 1.2.2

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{{(lim_{x \to 0} x)}^{2} \cdot lim_{x \to 0} e^{x}}{lim_{x \to 0} \tan^{2} (x)}$

Step 1.2.3

Move the limit into the exponent.

$\frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x}}{lim_{x \to 0} \tan^{2} (x)}$

Step 1.2.4

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 1.2.4.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0^{2} \cdot e^{lim_{x \to 0} x}}{lim_{x \to 0} \tan^{2} (x)}$

Step 1.2.4.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0^{2} \cdot e^{0}}{lim_{x \to 0} \tan^{2} (x)}$

Step 1.2.5

Simplify the answer.

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Step 1.2.5.1

Raising $0$ to any positive power yields $0$ .

$\frac{0 \cdot e^{0}}{lim_{x \to 0} \tan^{2} (x)}$

Step 1.2.5.2

Anything raised to $0$ is $1$ .

$\frac{0 \cdot 1}{lim_{x \to 0} \tan^{2} (x)}$

Step 1.2.5.3

Multiply $0$ by $1$ .

$\frac{0}{lim_{x \to 0} \tan^{2} (x)}$

Step 1.3

Evaluate the limit of the denominator.

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Step 1.3.1

Evaluate the limit.

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Step 1.3.1.1

Move the exponent $2$ from $\tan^{2} (x)$ outside the limit using the Limits Power Rule.

$\frac{0}{{(lim_{x \to 0} \tan (x))}^{2}}$

Step 1.3.1.2

Move the limit inside the trig function because tangent is continuous.

$\frac{0}{\tan^{2} (lim_{x \to 0} x)}$

Step 1.3.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{0}{\tan^{2} (0)}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

The exact value of $\tan (0)$ is $0$ .

$\frac{0}{0^{2}}$

Step 1.3.3.2

Raising $0$ to any positive power yields $0$ .

$\frac{0}{0}$

Step 1.3.3.3

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.3.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{0}{0}$

Step 2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{x^{2} e^{x}}{\tan^{2} (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [x^{2} e^{x}]}{\frac{d}{d x} [\tan^{2} (x)]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [x^{2} e^{x}]}{\frac{d}{d x} [\tan^{2} (x)]}$

Step 3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = e^{x}$ .

$lim_{x \to 0} \frac{x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [\tan^{2} (x)]}$

Step 3.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$lim_{x \to 0} \frac{x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [\tan^{2} (x)]}$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$lim_{x \to 0} \frac{x^{2} e^{x} + e^{x} (2 x)}{\frac{d}{d x} [\tan^{2} (x)]}$

Step 3.5

Simplify.

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Step 3.5.1

Reorder terms.

$lim_{x \to 0} \frac{e^{x} x^{2} + 2 e^{x} x}{\frac{d}{d x} [\tan^{2} (x)]}$

Step 3.5.2

Reorder factors in $e^{x} x^{2} + 2 e^{x} x$ .

$lim_{x \to 0} \frac{x^{2} e^{x} + 2 x e^{x}}{\frac{d}{d x} [\tan^{2} (x)]}$

Step 3.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \tan (x)$ .

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Step 3.6.1

To apply the Chain Rule, set $u$ as $\tan (x)$ .

$lim_{x \to 0} \frac{x^{2} e^{x} + 2 x e^{x}}{\frac{d}{d u} [u^{2}] \frac{d}{d x} [\tan (x)]}$

Step 3.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$lim_{x \to 0} \frac{x^{2} e^{x} + 2 x e^{x}}{2 u \frac{d}{d x} [\tan (x)]}$

Step 3.6.3

Replace all occurrences of $u$ with $\tan (x)$ .

$lim_{x \to 0} \frac{x^{2} e^{x} + 2 x e^{x}}{2 \tan (x) \frac{d}{d x} [\tan (x)]}$

Step 3.7

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$lim_{x \to 0} \frac{x^{2} e^{x} + 2 x e^{x}}{2 \tan (x) \sec^{2} (x)}$

Step 3.8

Reorder the factors of $2 \tan (x) \sec^{2} (x)$ .

$lim_{x \to 0} \frac{x^{2} e^{x} + 2 x e^{x}}{2 \sec^{2} (x) \tan (x)}$

Step 4

Move the term $\frac{1}{2}$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 2 x e^{x}}{\sec^{2} (x) \tan (x)}$

Step 5

Apply L'Hospital's rule.

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Step 5.1

Evaluate the limit of the numerator and the limit of the denominator.

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Step 5.1.1

Take the limit of the numerator and the limit of the denominator.

$\frac{1}{2} \cdot \frac{lim_{x \to 0} x^{2} e^{x} + 2 x e^{x}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2

Evaluate the limit of the numerator.

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Step 5.1.2.1

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{2} \cdot \frac{lim_{x \to 0} x^{2} e^{x} + lim_{x \to 0} 2 x e^{x}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.2

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{2} \cdot \frac{lim_{x \to 0} x^{2} \cdot lim_{x \to 0} e^{x} + lim_{x \to 0} 2 x e^{x}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.3

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot lim_{x \to 0} e^{x} + lim_{x \to 0} 2 x e^{x}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.4

Move the limit into the exponent.

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + lim_{x \to 0} 2 x e^{x}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.5

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 2 lim_{x \to 0} x e^{x}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.6

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 2 lim_{x \to 0} x \cdot lim_{x \to 0} e^{x}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.7

Move the limit into the exponent.

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 2 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.8

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 5.1.2.8.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{lim_{x \to 0} x} + 2 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.8.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{0} + 2 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.8.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{0} + 2 \cdot 0 \cdot e^{lim_{x \to 0} x}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.8.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{0} + 2 \cdot 0 \cdot e^{0}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.9

Simplify the answer.

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Step 5.1.2.9.1

Simplify each term.

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Step 5.1.2.9.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} \cdot \frac{0 \cdot e^{0} + 2 \cdot 0 \cdot e^{0}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.9.1.2

Anything raised to $0$ is $1$ .

$\frac{1}{2} \cdot \frac{0 \cdot 1 + 2 \cdot 0 \cdot e^{0}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.9.1.3

Multiply $0$ by $1$ .

$\frac{1}{2} \cdot \frac{0 + 2 \cdot 0 \cdot e^{0}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.9.1.4

Multiply $2$ by $0$ .

$\frac{1}{2} \cdot \frac{0 + 0 \cdot e^{0}}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.9.1.5

Anything raised to $0$ is $1$ .

$\frac{1}{2} \cdot \frac{0 + 0 \cdot 1}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.9.1.6

Multiply $0$ by $1$ .

$\frac{1}{2} \cdot \frac{0 + 0}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.2.9.2

Add $0$ and $0$ .

$\frac{1}{2} \cdot \frac{0}{lim_{x \to 0} \sec^{2} (x) \tan (x)}$

Step 5.1.3

Evaluate the limit of the denominator.

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Step 5.1.3.1

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{2} \cdot \frac{0}{lim_{x \to 0} \sec^{2} (x) \cdot lim_{x \to 0} \tan (x)}$

Step 5.1.3.2

Move the exponent $2$ from $\sec^{2} (x)$ outside the limit using the Limits Power Rule.

$\frac{1}{2} \cdot \frac{0}{{(lim_{x \to 0} \sec (x))}^{2} \cdot lim_{x \to 0} \tan (x)}$

Step 5.1.3.3

Move the limit inside the trig function because secant is continuous.

$\frac{1}{2} \cdot \frac{0}{\sec^{2} (lim_{x \to 0} x) \cdot lim_{x \to 0} \tan (x)}$

Step 5.1.3.4

Move the limit inside the trig function because tangent is continuous.

$\frac{1}{2} \cdot \frac{0}{\sec^{2} (lim_{x \to 0} x) \cdot \tan (lim_{x \to 0} x)}$

Step 5.1.3.5

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 5.1.3.5.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0}{\sec^{2} (0) \cdot \tan (lim_{x \to 0} x)}$

Step 5.1.3.5.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0}{\sec^{2} (0) \cdot \tan (0)}$

Step 5.1.3.6

Simplify the answer.

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Step 5.1.3.6.1

The exact value of $\sec (0)$ is $1$ .

$\frac{1}{2} \cdot \frac{0}{1^{2} \cdot \tan (0)}$

Step 5.1.3.6.2

One to any power is one.

$\frac{1}{2} \cdot \frac{0}{1 \cdot \tan (0)}$

Step 5.1.3.6.3

Multiply $\tan (0)$ by $1$ .

$\frac{1}{2} \cdot \frac{0}{\tan (0)}$

Step 5.1.3.6.4

The exact value of $\tan (0)$ is $0$ .

$\frac{1}{2} \cdot \frac{0}{0}$

Step 5.1.3.6.5

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 5.1.3.7

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 5.1.4

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{1}{2} \cdot \frac{0}{0}$

Step 5.2

Since $\frac{0}{0}$ is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

$lim_{x \to 0} \frac{x^{2} e^{x} + 2 x e^{x}}{\sec^{2} (x) \tan (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [x^{2} e^{x} + 2 x e^{x}]}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3

Find the derivative of the numerator and denominator.

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Step 5.3.1

Differentiate the numerator and denominator.

$\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [x^{2} e^{x} + 2 x e^{x}]}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.2

By the Sum Rule, the derivative of $x^{2} e^{x} + 2 x e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [2 x e^{x}]$ .

$\frac{1}{2} lim_{x \to 0} \frac{\frac{d}{d x} [x^{2} e^{x}] + \frac{d}{d x} [2 x e^{x}]}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.3

Evaluate $\frac{d}{d x} [x^{2} e^{x}]$ .

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Step 5.3.3.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = e^{x}$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x e^{x}]}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + e^{x} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x e^{x}]}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + e^{x} \cdot 2 x + \frac{d}{d x} [2 x e^{x}]}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.4

Evaluate $\frac{d}{d x} [2 x e^{x}]$ .

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Step 5.3.4.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x e^{x}$ with respect to $x$ is $2 \frac{d}{d x} [x e^{x}]$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + e^{x} \cdot 2 x + 2 \frac{d}{d x} [x e^{x}]}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.4.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{x}$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + e^{x} \cdot 2 x + 2 (x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x])}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.4.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + e^{x} \cdot 2 x + 2 (x e^{x} + e^{x} \frac{d}{d x} [x])}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + e^{x} \cdot 2 x + 2 (x e^{x} + e^{x} \cdot 1)}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.4.5

Multiply $e^{x}$ by $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + e^{x} \cdot 2 x + 2 (x e^{x} + e^{x})}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.5

Simplify.

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Step 5.3.5.1

Apply the distributive property.

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + e^{x} \cdot 2 x + 2 x e^{x} + 2 e^{x}}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.5.2

Add $e^{x} (2 x)$ and $2 x e^{x}$ .

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Step 5.3.5.2.1

Move $e^{x}$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 2 x e^{x} + 2 x e^{x} + 2 e^{x}}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.5.2.2

Add $2 x e^{x}$ and $2 x e^{x}$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.5.3

Reorder terms.

$\frac{1}{2} lim_{x \to 0} \frac{e^{x} x^{2} + 4 e^{x} x + 2 e^{x}}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.5.4

Reorder factors in $e^{x} x^{2} + 4 e^{x} x + 2 e^{x}$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\frac{d}{d x} [\sec^{2} (x) \tan (x)]}$

Step 5.3.6

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = \sec^{2} (x)$ and $g (x) = \tan (x)$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{2} (x) \frac{d}{d x} [\tan (x)] + \tan (x) \frac{d}{d x} [\sec^{2} (x)]}$

Step 5.3.7

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{2} (x) \sec^{2} (x) + \tan (x) \frac{d}{d x} [\sec^{2} (x)]}$

Step 5.3.8

Multiply $\sec^{2} (x)$ by $\sec^{2} (x)$ by adding the exponents.

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Step 5.3.8.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{{\sec (x)}^{2 + 2} + \tan (x) \frac{d}{d x} [\sec^{2} (x)]}$

Step 5.3.8.2

Add $2$ and $2$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + \tan (x) \frac{d}{d x} [\sec^{2} (x)]}$

Step 5.3.9

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = \sec (x)$ .

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Step 5.3.9.1

To apply the Chain Rule, set $u$ as $\sec (x)$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + \tan (x) \frac{d}{d u} [u^{2}] \frac{d}{d x} [\sec (x)]}$

Step 5.3.9.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + \tan (x) \cdot 2 u \frac{d}{d x} [\sec (x)]}$

Step 5.3.9.3

Replace all occurrences of $u$ with $\sec (x)$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + \tan (x) \cdot 2 \sec (x) \frac{d}{d x} [\sec (x)]}$

Step 5.3.10

Move $2$ to the left of $\tan (x)$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + 2 \cdot \tan (x) \sec (x) \frac{d}{d x} [\sec (x)]}$

Step 5.3.11

The derivative of $\sec (x)$ with respect to $x$ is $\sec (x) \tan (x)$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + 2 \tan (x) \sec (x) \sec (x) \tan (x)}$

Step 5.3.12

Raise $\tan (x)$ to the power of $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + 2 \tan^{1} (x) \tan (x) \sec (x) \sec (x)}$

Step 5.3.13

Raise $\tan (x)$ to the power of $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + 2 \tan^{1} (x) \tan^{1} (x) \sec (x) \sec (x)}$

Step 5.3.14

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + 2 {\tan (x)}^{1 + 1} \sec (x) \sec (x)}$

Step 5.3.15

Add $1$ and $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + 2 \tan^{2} (x) \sec (x) \sec (x)}$

Step 5.3.16

Raise $\sec (x)$ to the power of $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + 2 \tan^{2} (x) \sec^{1} (x) \sec (x)}$

Step 5.3.17

Raise $\sec (x)$ to the power of $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + 2 \tan^{2} (x) \sec^{1} (x) \sec^{1} (x)}$

Step 5.3.18

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + 2 \tan^{2} (x) {\sec (x)}^{1 + 1}}$

Step 5.3.19

Add $1$ and $1$ .

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{\sec^{4} (x) + 2 \tan^{2} (x) \sec^{2} (x)}$

Step 5.3.20

Reorder terms.

$\frac{1}{2} lim_{x \to 0} \frac{x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{2 \sec^{2} (x) \tan^{2} (x) + \sec^{4} (x)}$

Step 6

Evaluate the limit.

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Step 6.1

Split the limit using the Limits Quotient Rule on the limit as $x$ approaches $0$ .

$\frac{1}{2} \cdot \frac{lim_{x \to 0} x^{2} e^{x} + 4 x e^{x} + 2 e^{x}}{lim_{x \to 0} 2 \sec^{2} (x) \tan^{2} (x) + \sec^{4} (x)}$

Step 6.2

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{2} \cdot \frac{lim_{x \to 0} x^{2} e^{x} + lim_{x \to 0} 4 x e^{x} + lim_{x \to 0} 2 e^{x}}{lim_{x \to 0} 2 \sec^{2} (x) \tan^{2} (x) + \sec^{4} (x)}$

Step 6.3

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{2} \cdot \frac{lim_{x \to 0} x^{2} \cdot lim_{x \to 0} e^{x} + lim_{x \to 0} 4 x e^{x} + lim_{x \to 0} 2 e^{x}}{lim_{x \to 0} 2 \sec^{2} (x) \tan^{2} (x) + \sec^{4} (x)}$

Step 6.4

Move the exponent $2$ from $x^{2}$ outside the limit using the Limits Power Rule.

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot lim_{x \to 0} e^{x} + lim_{x \to 0} 4 x e^{x} + lim_{x \to 0} 2 e^{x}}{lim_{x \to 0} 2 \sec^{2} (x) \tan^{2} (x) + \sec^{4} (x)}$

Step 6.5

Move the limit into the exponent.

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + lim_{x \to 0} 4 x e^{x} + lim_{x \to 0} 2 e^{x}}{lim_{x \to 0} 2 \sec^{2} (x) \tan^{2} (x) + \sec^{4} (x)}$

Step 6.6

Move the term $4$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x e^{x} + lim_{x \to 0} 2 e^{x}}{lim_{x \to 0} 2 \sec^{2} (x) \tan^{2} (x) + \sec^{4} (x)}$

Step 6.7

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x \cdot lim_{x \to 0} e^{x} + lim_{x \to 0} 2 e^{x}}{lim_{x \to 0} 2 \sec^{2} (x) \tan^{2} (x) + \sec^{4} (x)}$

Step 6.8

Move the limit into the exponent.

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x} + lim_{x \to 0} 2 e^{x}}{lim_{x \to 0} 2 \sec^{2} (x) \tan^{2} (x) + \sec^{4} (x)}$

Step 6.9

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x} + 2 lim_{x \to 0} e^{x}}{lim_{x \to 0} 2 \sec^{2} (x) \tan^{2} (x) + \sec^{4} (x)}$

Step 6.10

Move the limit into the exponent.

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x} + 2 e^{lim_{x \to 0} x}}{lim_{x \to 0} 2 \sec^{2} (x) \tan^{2} (x) + \sec^{4} (x)}$

Step 6.11

Split the limit using the Sum of Limits Rule on the limit as $x$ approaches $0$ .

Step 6.12

Move the term $2$ outside of the limit because it is constant with respect to $x$ .

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x} + 2 e^{lim_{x \to 0} x}}{2 lim_{x \to 0} \sec^{2} (x) \tan^{2} (x) + lim_{x \to 0} \sec^{4} (x)}$

Step 6.13

Split the limit using the Product of Limits Rule on the limit as $x$ approaches $0$ .

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x} + 2 e^{lim_{x \to 0} x}}{2 lim_{x \to 0} \sec^{2} (x) \cdot lim_{x \to 0} \tan^{2} (x) + lim_{x \to 0} \sec^{4} (x)}$

Step 6.14

Move the exponent $2$ from $\sec^{2} (x)$ outside the limit using the Limits Power Rule.

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x} + 2 e^{lim_{x \to 0} x}}{2 {(lim_{x \to 0} \sec (x))}^{2} \cdot lim_{x \to 0} \tan^{2} (x) + lim_{x \to 0} \sec^{4} (x)}$

Step 6.15

Move the limit inside the trig function because secant is continuous.

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x} + 2 e^{lim_{x \to 0} x}}{2 \sec^{2} (lim_{x \to 0} x) \cdot lim_{x \to 0} \tan^{2} (x) + lim_{x \to 0} \sec^{4} (x)}$

Step 6.16

Move the exponent $2$ from $\tan^{2} (x)$ outside the limit using the Limits Power Rule.

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x} + 2 e^{lim_{x \to 0} x}}{2 \sec^{2} (lim_{x \to 0} x) \cdot {(lim_{x \to 0} \tan (x))}^{2} + lim_{x \to 0} \sec^{4} (x)}$

Step 6.17

Move the limit inside the trig function because tangent is continuous.

$\frac{1}{2} \cdot \frac{{(lim_{x \to 0} x)}^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x} + 2 e^{lim_{x \to 0} x}}{2 \sec^{2} (lim_{x \to 0} x) \cdot \tan^{2} (lim_{x \to 0} x) + lim_{x \to 0} \sec^{4} (x)}$

Step 6.18

Move the exponent $4$ from $\sec^{4} (x)$ outside the limit using the Limits Power Rule.

Step 6.19

Move the limit inside the trig function because secant is continuous.

Step 7

Evaluate the limits by plugging in $0$ for all occurrences of $x$ .

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Step 7.1

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{lim_{x \to 0} x} + 4 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x} + 2 e^{lim_{x \to 0} x}}{2 \sec^{2} (lim_{x \to 0} x) \cdot \tan^{2} (lim_{x \to 0} x) + \sec^{4} (lim_{x \to 0} x)}$

Step 7.2

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{0} + 4 lim_{x \to 0} x \cdot e^{lim_{x \to 0} x} + 2 e^{lim_{x \to 0} x}}{2 \sec^{2} (lim_{x \to 0} x) \cdot \tan^{2} (lim_{x \to 0} x) + \sec^{4} (lim_{x \to 0} x)}$

Step 7.3

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{0} + 4 \cdot 0 \cdot e^{lim_{x \to 0} x} + 2 e^{lim_{x \to 0} x}}{2 \sec^{2} (lim_{x \to 0} x) \cdot \tan^{2} (lim_{x \to 0} x) + \sec^{4} (lim_{x \to 0} x)}$

Step 7.4

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{0} + 4 \cdot 0 \cdot e^{0} + 2 e^{lim_{x \to 0} x}}{2 \sec^{2} (lim_{x \to 0} x) \cdot \tan^{2} (lim_{x \to 0} x) + \sec^{4} (lim_{x \to 0} x)}$

Step 7.5

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{0} + 4 \cdot 0 \cdot e^{0} + 2 e^{0}}{2 \sec^{2} (lim_{x \to 0} x) \cdot \tan^{2} (lim_{x \to 0} x) + \sec^{4} (lim_{x \to 0} x)}$

Step 7.6

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{0} + 4 \cdot 0 \cdot e^{0} + 2 e^{0}}{2 \sec^{2} (0) \cdot \tan^{2} (lim_{x \to 0} x) + \sec^{4} (lim_{x \to 0} x)}$

Step 7.7

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{0} + 4 \cdot 0 \cdot e^{0} + 2 e^{0}}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (lim_{x \to 0} x)}$

Step 7.8

Evaluate the limit of $x$ by plugging in $0$ for $x$ .

$\frac{1}{2} \cdot \frac{0^{2} \cdot e^{0} + 4 \cdot 0 \cdot e^{0} + 2 e^{0}}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8

Simplify the answer.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} \cdot \frac{0 \cdot e^{0} + 4 \cdot 0 \cdot e^{0} + 2 e^{0}}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.1.2

Anything raised to $0$ is $1$ .

$\frac{1}{2} \cdot \frac{0 \cdot 1 + 4 \cdot 0 \cdot e^{0} + 2 e^{0}}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.1.3

Multiply $0$ by $1$ .

$\frac{1}{2} \cdot \frac{0 + 4 \cdot 0 \cdot e^{0} + 2 e^{0}}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.1.4

Multiply $4$ by $0$ .

$\frac{1}{2} \cdot \frac{0 + 0 \cdot e^{0} + 2 e^{0}}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.1.5

Anything raised to $0$ is $1$ .

$\frac{1}{2} \cdot \frac{0 + 0 \cdot 1 + 2 e^{0}}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.1.6

Multiply $0$ by $1$ .

$\frac{1}{2} \cdot \frac{0 + 0 + 2 e^{0}}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.1.7

Anything raised to $0$ is $1$ .

$\frac{1}{2} \cdot \frac{0 + 0 + 2 \cdot 1}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.1.8

Multiply $2$ by $1$ .

$\frac{1}{2} \cdot \frac{0 + 0 + 2}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.1.9

Add $0$ and $0$ .

$\frac{1}{2} \cdot \frac{0 + 2}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.1.10

Add $0$ and $2$ .

$\frac{1}{2} \cdot \frac{2}{2 \sec^{2} (0) \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.2

Simplify the denominator.

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Step 8.2.1

The exact value of $\sec (0)$ is $1$ .

$\frac{1}{2} \cdot \frac{2}{2 \cdot 1^{2} \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.2.2

One to any power is one.

$\frac{1}{2} \cdot \frac{2}{2 \cdot 1 \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.2.3

Multiply $2$ by $1$ .

$\frac{1}{2} \cdot \frac{2}{2 \cdot \tan^{2} (0) + \sec^{4} (0)}$

Step 8.2.4

The exact value of $\tan (0)$ is $0$ .

$\frac{1}{2} \cdot \frac{2}{2 \cdot 0^{2} + \sec^{4} (0)}$

Step 8.2.5

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} \cdot \frac{2}{2 \cdot 0 + \sec^{4} (0)}$

Step 8.2.6

Multiply $2$ by $0$ .

$\frac{1}{2} \cdot \frac{2}{0 + \sec^{4} (0)}$

Step 8.2.7

The exact value of $\sec (0)$ is $1$ .

$\frac{1}{2} \cdot \frac{2}{0 + 1^{4}}$

Step 8.2.8

One to any power is one.

$\frac{1}{2} \cdot \frac{2}{0 + 1}$

Step 8.2.9

Add $0$ and $1$ .

$\frac{1}{2} \cdot \frac{2}{1}$

Step 8.3

Cancel the common factor of $2$ .

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Step 8.3.1

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2}{1}$

Step 8.3.2

Rewrite the expression.

$1$