Calculus Examples

$lim_{x \to 0} \frac{9 x^{2}}{\ln (\sec (x))}$

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.

$\frac{lim_{x \to 0} 9 x^{2}}{lim_{x \to 0} \ln (\sec (x))}$

Step 1.2

Evaluate the limit of the numerator.

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Step 1.2.1

Evaluate the limit.

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Step 1.2.1.1

Move the term $9$ outside of the limit because it is constant with respect to $x$ .

$\frac{9 lim_{x \to 0} x^{2}}{lim_{x \to 0} \ln (\sec (x))}$

Step 1.2.1.2

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

$\frac{9 {(lim_{x \to 0} x)}^{2}}{lim_{x \to 0} \ln (\sec (x))}$

Step 1.2.2

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

$\frac{9 \cdot 0^{2}}{lim_{x \to 0} \ln (\sec (x))}$

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

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

$\frac{9 \cdot 0}{lim_{x \to 0} \ln (\sec (x))}$

Step 1.2.3.2

Multiply $9$ by $0$ .

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

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 limit inside the logarithm.

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

Step 1.3.1.2

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

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

Step 1.3.2

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

$\frac{0}{\ln (\sec (0))}$

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

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

$\frac{0}{\ln (1)}$

Step 1.3.3.2

The natural logarithm of $1$ is $0$ .

$\frac{0}{0}$

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.

$lim_{x \to 0} \frac{9 x^{2}}{\ln (\sec (x))} = lim_{x \to 0} \frac{\frac{d}{d x} [9 x^{2}]}{\frac{d}{d x} [\ln (\sec (x))]}$

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

$lim_{x \to 0} \frac{\frac{d}{d x} [9 x^{2}]}{\frac{d}{d x} [\ln (\sec (x))]}$

Step 3.2

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$lim_{x \to 0} \frac{9 \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [\ln (\sec (x))]}$

Step 3.3

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

$lim_{x \to 0} \frac{9 (2 x)}{\frac{d}{d x} [\ln (\sec (x))]}$

Step 3.4

Multiply $2$ by $9$ .

$lim_{x \to 0} \frac{18 x}{\frac{d}{d x} [\ln (\sec (x))]}$

Step 3.5

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

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Step 3.5.1

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

$lim_{x \to 0} \frac{18 x}{\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\sec (x)]}$

Step 3.5.2

The derivative of $\ln (u)$ with respect to $u$ is $\frac{1}{u}$ .

$lim_{x \to 0} \frac{18 x}{\frac{1}{u} \frac{d}{d x} [\sec (x)]}$

Step 3.5.3

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

$lim_{x \to 0} \frac{18 x}{\frac{1}{\sec (x)} \frac{d}{d x} [\sec (x)]}$

Step 3.6

Rewrite $\sec (x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{18 x}{\frac{1}{\frac{1}{\cos (x)}} \frac{d}{d x} [\sec (x)]}$

Step 3.7

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

$lim_{x \to 0} \frac{18 x}{1 \cos (x) \frac{d}{d x} [\sec (x)]}$

Step 3.8

Multiply $\cos (x)$ by $1$ .

$lim_{x \to 0} \frac{18 x}{\cos (x) \frac{d}{d x} [\sec (x)]}$

Step 3.9

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

$lim_{x \to 0} \frac{18 x}{\cos (x) (\sec (x) \tan (x))}$

Step 3.10

Remove parentheses.

$lim_{x \to 0} \frac{18 x}{\cos (x) \sec (x) \tan (x)}$

Step 3.11

Simplify.

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Step 3.11.1

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 3.11.1.1

Reorder $\cos (x)$ and $\sec (x)$ .

$lim_{x \to 0} \frac{18 x}{\sec (x) \cos (x) \tan (x)}$

Step 3.11.1.2

Rewrite $\cos (x) \sec (x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{18 x}{\frac{1}{\cos (x)} \cos (x) \tan (x)}$

Step 3.11.1.3

Cancel the common factors.

$lim_{x \to 0} \frac{18 x}{1 \tan (x)}$

Step 3.11.2

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

$lim_{x \to 0} \frac{18 x}{\tan (x)}$

Step 3.11.3

Rewrite $\tan (x)$ in terms of sines and cosines.

$lim_{x \to 0} \frac{18 x}{\frac{\sin (x)}{\cos (x)}}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0} 18 x \frac{\cos (x)}{\sin (x)}$

Step 5

Combine factors.

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Step 5.1

Combine $18$ and $\frac{\cos (x)}{\sin (x)}$ .

$lim_{x \to 0} x \frac{18 \cos (x)}{\sin (x)}$

Step 5.2

Combine $x$ and $\frac{18 \cos (x)}{\sin (x)}$ .

$lim_{x \to 0} \frac{x (18 \cos (x))}{\sin (x)}$

Step 6

Move the term $18$ outside of the limit because it is constant with respect to $x$ .

$18 lim_{x \to 0} \frac{x (\cos (x))}{\sin (x)}$

Step 7

Apply L'Hospital's rule.

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Step 7.1

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

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Step 7.1.1

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

$18 \frac{lim_{x \to 0} x (\cos (x))}{lim_{x \to 0} \sin (x)}$

Step 7.1.2

Evaluate the limit of the numerator.

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Step 7.1.2.1

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

$18 \frac{lim_{x \to 0} x \cdot lim_{x \to 0} \cos (x)}{lim_{x \to 0} \sin (x)}$

Step 7.1.2.2

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

$18 \frac{lim_{x \to 0} x \cdot \cos (lim_{x \to 0} x)}{lim_{x \to 0} \sin (x)}$

Step 7.1.2.3

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

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Step 7.1.2.3.1

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

$18 \frac{0 \cdot \cos (lim_{x \to 0} x)}{lim_{x \to 0} \sin (x)}$

Step 7.1.2.3.2

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

$18 \frac{0 \cdot \cos (0)}{lim_{x \to 0} \sin (x)}$

Step 7.1.2.4

Simplify the answer.

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Step 7.1.2.4.1

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

$18 \frac{0 \cdot 1}{lim_{x \to 0} \sin (x)}$

Step 7.1.2.4.2

Multiply $0$ by $1$ .

$18 \frac{0}{lim_{x \to 0} \sin (x)}$

Step 7.1.3

Evaluate the limit of the denominator.

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Step 7.1.3.1

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

$18 \frac{0}{\sin (lim_{x \to 0} x)}$

Step 7.1.3.2

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

$18 \frac{0}{\sin (0)}$

Step 7.1.3.3

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

$18 (\frac{0}{0})$

Step 7.1.3.4

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

Undefined

$18 (\frac{0}{0})$

Step 7.1.4

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

Undefined

$18 (\frac{0}{0})$

Step 7.2

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.

$lim_{x \to 0} \frac{x (\cos (x))}{\sin (x)} = lim_{x \to 0} \frac{\frac{d}{d x} [x (\cos (x))]}{\frac{d}{d x} [\sin (x)]}$

Step 7.3

Find the derivative of the numerator and denominator.

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Step 7.3.1

Differentiate the numerator and denominator.

$18 lim_{x \to 0} \frac{\frac{d}{d x} [x (\cos (x))]}{\frac{d}{d x} [\sin (x)]}$

Step 7.3.2

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 $f (x) = x$ and $g (x) = \cos (x)$ .

$18 lim_{x \to 0} \frac{x \frac{d}{d x} [\cos (x)] + \cos (x) \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (x)]}$

Step 7.3.3

The derivative of $\cos (x)$ with respect to $x$ is $- \sin (x)$ .

$18 lim_{x \to 0} \frac{x \cdot - 1 \sin (x) + \cos (x) \frac{d}{d x} [x]}{\frac{d}{d x} [\sin (x)]}$

Step 7.3.4

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

$18 lim_{x \to 0} \frac{x \cdot - 1 \sin (x) + \cos (x) \cdot 1}{\frac{d}{d x} [\sin (x)]}$

Step 7.3.5

Multiply $\cos (x)$ by $1$ .

$18 lim_{x \to 0} \frac{x \cdot - 1 \sin (x) + \cos (x)}{\frac{d}{d x} [\sin (x)]}$

Step 7.3.6

Reorder terms.

$18 lim_{x \to 0} \frac{- x \sin (x) + \cos (x)}{\frac{d}{d x} [\sin (x)]}$

Step 7.3.7

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$18 lim_{x \to 0} \frac{- x \sin (x) + \cos (x)}{\cos (x)}$

Step 8

Evaluate the limit.

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Step 8.1

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

$18 \frac{lim_{x \to 0} - x \sin (x) + \cos (x)}{lim_{x \to 0} \cos (x)}$

Step 8.2

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

$18 \frac{- lim_{x \to 0} x \sin (x) + lim_{x \to 0} \cos (x)}{lim_{x \to 0} \cos (x)}$

Step 8.3

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

$18 \frac{- lim_{x \to 0} x \cdot lim_{x \to 0} \sin (x) + lim_{x \to 0} \cos (x)}{lim_{x \to 0} \cos (x)}$

Step 8.4

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

$18 \frac{- lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + lim_{x \to 0} \cos (x)}{lim_{x \to 0} \cos (x)}$

Step 8.5

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

$18 \frac{- lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + \cos (lim_{x \to 0} x)}{lim_{x \to 0} \cos (x)}$

Step 8.6

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

$18 \frac{- lim_{x \to 0} x \cdot \sin (lim_{x \to 0} x) + \cos (lim_{x \to 0} x)}{\cos (lim_{x \to 0} x)}$

Step 9

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

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Step 9.1

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

$18 \frac{0 \cdot \sin (lim_{x \to 0} x) + \cos (lim_{x \to 0} x)}{\cos (lim_{x \to 0} x)}$

Step 9.2

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

$18 \frac{0 \cdot \sin (0) + \cos (lim_{x \to 0} x)}{\cos (lim_{x \to 0} x)}$

Step 9.3

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

$18 \frac{0 \cdot \sin (0) + \cos (0)}{\cos (lim_{x \to 0} x)}$

Step 9.4

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

$18 \frac{0 \cdot \sin (0) + \cos (0)}{\cos (0)}$

Step 10

Simplify the answer.

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Step 10.1

Simplify the numerator.

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Step 10.1.1

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

$18 \frac{0 \cdot 0 + \cos (0)}{\cos (0)}$

Step 10.1.2

Multiply $0$ by $0$ .

$18 \frac{0 + \cos (0)}{\cos (0)}$

Step 10.1.3

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

$18 \frac{0 + 1}{\cos (0)}$

Step 10.1.4

Add $0$ and $1$ .

$18 \frac{1}{\cos (0)}$

Step 10.2

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

$18 (\frac{1}{1})$

Step 10.3

Cancel the common factor of $1$ .

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Step 10.3.1

Cancel the common factor.

$18 (\frac{1}{1})$

Step 10.3.2

Rewrite the expression.

$18 \cdot 1$

Step 10.4

Multiply $18$ by $1$ .

$18$