Calculus Examples

$lim_{x \to 0} \frac{8 \sec (x) - 8}{x^{2}}$

Step 1

Apply L'Hospital's rule.

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Step 1.1

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

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Step 1.1.1

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

$\frac{lim_{x \to 0} 8 \sec (x) - 8}{lim_{x \to 0} x^{2}}$

Step 1.1.2

Evaluate the limit of the numerator.

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Step 1.1.2.1

Evaluate the limit.

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Step 1.1.2.1.1

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

$\frac{lim_{x \to 0} 8 \sec (x) - lim_{x \to 0} 8}{lim_{x \to 0} x^{2}}$

Step 1.1.2.1.2

Move the term $8$ outside of the limit because it is constant with respect to $x$ .

$\frac{8 lim_{x \to 0} \sec (x) - lim_{x \to 0} 8}{lim_{x \to 0} x^{2}}$

Step 1.1.2.1.3

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

$\frac{8 \sec (lim_{x \to 0} x) - lim_{x \to 0} 8}{lim_{x \to 0} x^{2}}$

Step 1.1.2.1.4

Evaluate the limit of $8$ which is constant as $x$ approaches $0$ .

$\frac{8 \sec (lim_{x \to 0} x) - 1 \cdot 8}{lim_{x \to 0} x^{2}}$

Step 1.1.2.2

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

$\frac{8 \sec (0) - 1 \cdot 8}{lim_{x \to 0} x^{2}}$

Step 1.1.2.3

Simplify the answer.

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Step 1.1.2.3.1

Simplify each term.

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Step 1.1.2.3.1.1

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

$\frac{8 \cdot 1 - 1 \cdot 8}{lim_{x \to 0} x^{2}}$

Step 1.1.2.3.1.2

Multiply $8$ by $1$ .

$\frac{8 - 1 \cdot 8}{lim_{x \to 0} x^{2}}$

Step 1.1.2.3.1.3

Multiply $- 1$ by $8$ .

$\frac{8 - 8}{lim_{x \to 0} x^{2}}$

Step 1.1.2.3.2

Subtract $8$ from $8$ .

$\frac{0}{lim_{x \to 0} x^{2}}$

Step 1.1.3

Evaluate the limit of the denominator.

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

$\frac{0}{0}$

Step 1.1.3.4

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

Undefined

$\frac{0}{0}$

Step 1.1.4

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

Undefined

$\frac{0}{0}$

Step 1.2

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.

$lim_{x \to 0} \frac{8 \sec (x) - 8}{x^{2}} = lim_{x \to 0} \frac{\frac{d}{d x} [8 \sec (x) - 8]}{\frac{d}{d x} [x^{2}]}$

Step 1.3

Find the derivative of the numerator and denominator.

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Step 1.3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [8 \sec (x) - 8]}{\frac{d}{d x} [x^{2}]}$

Step 1.3.2

By the Sum Rule, the derivative of $8 \sec (x) - 8$ with respect to $x$ is $\frac{d}{d x} [8 \sec (x)] + \frac{d}{d x} [- 8]$ .

$lim_{x \to 0} \frac{\frac{d}{d x} [8 \sec (x)] + \frac{d}{d x} [- 8]}{\frac{d}{d x} [x^{2}]}$

Step 1.3.3

Evaluate $\frac{d}{d x} [8 \sec (x)]$ .

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Step 1.3.3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 \sec (x)$ with respect to $x$ is $8 \frac{d}{d x} [\sec (x)]$ .

$lim_{x \to 0} \frac{8 \frac{d}{d x} [\sec (x)] + \frac{d}{d x} [- 8]}{\frac{d}{d x} [x^{2}]}$

Step 1.3.3.2

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

$lim_{x \to 0} \frac{8 \sec (x) \tan (x) + \frac{d}{d x} [- 8]}{\frac{d}{d x} [x^{2}]}$

Step 1.3.4

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$lim_{x \to 0} \frac{8 \sec (x) \tan (x) + 0}{\frac{d}{d x} [x^{2}]}$

Step 1.3.5

Add $8 \sec (x) \tan (x)$ and $0$ .

$lim_{x \to 0} \frac{8 \sec (x) \tan (x)}{\frac{d}{d x} [x^{2}]}$

Step 1.3.6

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

$lim_{x \to 0} \frac{8 \sec (x) \tan (x)}{2 x}$

Step 1.4

Cancel the common factor of $8$ and $2$ .

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Step 1.4.1

Factor $2$ out of $8 \sec (x) \tan (x)$ .

$lim_{x \to 0} \frac{2 (4 \sec (x) \tan (x))}{2 x}$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $2 x$ .

$lim_{x \to 0} \frac{2 (4 \sec (x) \tan (x))}{2 (x)}$

Step 1.4.2.2

Cancel the common factor.

$lim_{x \to 0} \frac{2 (4 \sec (x) \tan (x))}{2 x}$

Step 1.4.2.3

Rewrite the expression.

$lim_{x \to 0} \frac{4 \sec (x) \tan (x)}{x}$

Step 2

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

$4 lim_{x \to 0} \frac{\sec (x) \tan (x)}{x}$

Step 3

Apply L'Hospital's rule.

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Step 3.1

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

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Step 3.1.1

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

$4 \frac{lim_{x \to 0} \sec (x) \tan (x)}{lim_{x \to 0} x}$

Step 3.1.2

Evaluate the limit of the numerator.

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Step 3.1.2.1

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

$4 \frac{lim_{x \to 0} \sec (x) \cdot lim_{x \to 0} \tan (x)}{lim_{x \to 0} x}$

Step 3.1.2.2

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

$4 \frac{\sec (lim_{x \to 0} x) \cdot lim_{x \to 0} \tan (x)}{lim_{x \to 0} x}$

Step 3.1.2.3

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

$4 \frac{\sec (lim_{x \to 0} x) \cdot \tan (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 3.1.2.4

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

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Step 3.1.2.4.1

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

$4 \frac{\sec (0) \cdot \tan (lim_{x \to 0} x)}{lim_{x \to 0} x}$

Step 3.1.2.4.2

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

$4 \frac{\sec (0) \cdot \tan (0)}{lim_{x \to 0} x}$

Step 3.1.2.5

Simplify the answer.

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Step 3.1.2.5.1

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

$4 \frac{1 \cdot \tan (0)}{lim_{x \to 0} x}$

Step 3.1.2.5.2

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

$4 \frac{\tan (0)}{lim_{x \to 0} x}$

Step 3.1.2.5.3

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

$4 \frac{0}{lim_{x \to 0} x}$

Step 3.1.3

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

$4 (\frac{0}{0})$

Step 3.1.4

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

Undefined

$4 (\frac{0}{0})$

Step 3.2

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.

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

Step 3.3

Find the derivative of the numerator and denominator.

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Step 3.3.1

Differentiate the numerator and denominator.

$4 lim_{x \to 0} \frac{\frac{d}{d x} [\sec (x) \tan (x)]}{\frac{d}{d x} [x]}$

Step 3.3.2

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 $f (x) = \sec (x)$ and $g (x) = \tan (x)$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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Step 3.3.4.1

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

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Step 3.3.4.1.1

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

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

Step 3.3.4.1.2

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

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

Step 3.3.4.2

Add $1$ and $2$ .

$4 lim_{x \to 0} \frac{\sec^{3} (x) + \tan (x) \frac{d}{d x} [\sec (x)]}{\frac{d}{d x} [x]}$

Step 3.3.5

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

$4 lim_{x \to 0} \frac{\sec^{3} (x) + \tan (x) \sec (x) \tan (x)}{\frac{d}{d x} [x]}$

Step 3.3.6

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

$4 lim_{x \to 0} \frac{\sec^{3} (x) + \tan^{1} (x) \tan (x) \sec (x)}{\frac{d}{d x} [x]}$

Step 3.3.7

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

$4 lim_{x \to 0} \frac{\sec^{3} (x) + \tan^{1} (x) \tan^{1} (x) \sec (x)}{\frac{d}{d x} [x]}$

Step 3.3.8

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

$4 lim_{x \to 0} \frac{\sec^{3} (x) + {\tan (x)}^{1 + 1} \sec (x)}{\frac{d}{d x} [x]}$

Step 3.3.9

Add $1$ and $1$ .

$4 lim_{x \to 0} \frac{\sec^{3} (x) + \tan^{2} (x) \sec (x)}{\frac{d}{d x} [x]}$

Step 3.3.10

Reorder terms.

$4 lim_{x \to 0} \frac{\tan^{2} (x) \sec (x) + \sec^{3} (x)}{\frac{d}{d x} [x]}$

Step 3.3.11

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

$4 lim_{x \to 0} \frac{\tan^{2} (x) \sec (x) + \sec^{3} (x)}{1}$

Step 3.4

Divide $\tan^{2} (x) \sec (x) + \sec^{3} (x)$ by $1$ .

$4 lim_{x \to 0} \tan^{2} (x) \sec (x) + \sec^{3} (x)$

Step 4

Evaluate the limit.

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Step 4.1

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

$4 (lim_{x \to 0} \tan^{2} (x) \sec (x) + lim_{x \to 0} \sec^{3} (x))$

Step 4.2

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

$4 (lim_{x \to 0} \tan^{2} (x) \cdot lim_{x \to 0} \sec (x) + lim_{x \to 0} \sec^{3} (x))$

Step 4.3

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

$4 ({(lim_{x \to 0} \tan (x))}^{2} \cdot lim_{x \to 0} \sec (x) + lim_{x \to 0} \sec^{3} (x))$

Step 4.4

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

$4 (\tan^{2} (lim_{x \to 0} x) \cdot lim_{x \to 0} \sec (x) + lim_{x \to 0} \sec^{3} (x))$

Step 4.5

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

$4 (\tan^{2} (lim_{x \to 0} x) \cdot \sec (lim_{x \to 0} x) + lim_{x \to 0} \sec^{3} (x))$

Step 4.6

Move the exponent $3$ from $\sec^{3} (x)$ outside the limit using the Limits Power Rule.

$4 (\tan^{2} (lim_{x \to 0} x) \cdot \sec (lim_{x \to 0} x) + {(lim_{x \to 0} \sec (x))}^{3})$

Step 4.7

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

$4 (\tan^{2} (lim_{x \to 0} x) \cdot \sec (lim_{x \to 0} x) + \sec^{3} (lim_{x \to 0} x))$

Step 5

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

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Step 5.1

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

$4 (\tan^{2} (0) \cdot \sec (lim_{x \to 0} x) + \sec^{3} (lim_{x \to 0} x))$

Step 5.2

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

$4 (\tan^{2} (0) \cdot \sec (0) + \sec^{3} (lim_{x \to 0} x))$

Step 5.3

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

$4 (\tan^{2} (0) \cdot \sec (0) + \sec^{3} (0))$

Step 6

Simplify the answer.

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Step 6.1

Simplify each term.

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Step 6.1.1

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

$4 (0^{2} \cdot \sec (0) + \sec^{3} (0))$

Step 6.1.2

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

$4 (0 \cdot \sec (0) + \sec^{3} (0))$

Step 6.1.3

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

$4 (0 \cdot 1 + \sec^{3} (0))$

Step 6.1.4

Multiply $0$ by $1$ .

$4 (0 + \sec^{3} (0))$

Step 6.1.5

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

$4 (0 + 1^{3})$

Step 6.1.6

One to any power is one.

$4 (0 + 1)$

Step 6.2

Add $0$ and $1$ .

$4 \cdot 1$

Step 6.3

Multiply $4$ by $1$ .

$4$