Calculus Examples

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

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.

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

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 limit inside the trig function because sine is continuous.

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

Step 1.2.1.2

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

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

Step 1.2.2

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

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

Step 1.2.3

Simplify the answer.

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Step 1.2.3.1

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

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

Step 1.2.3.2

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

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

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.

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

Step 1.3.1.2

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

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

Step 1.3.2

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

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

Step 1.3.3

Simplify the answer.

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Step 1.3.3.1

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

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

Step 3

Find the derivative of the numerator and denominator.

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Step 3.1

Differentiate the numerator and denominator.

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

Step 3.2

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

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x^{2}$ .

$lim_{x \to 0} \frac{\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x^{2}]}{\frac{d}{d x} [\ln (\cos (x))]}$

Step 3.2.2

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

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

Step 3.2.3

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 3.3

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

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

Step 3.4

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

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

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

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Step 3.5.1

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

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

Step 3.5.2

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

$lim_{x \to 0} \frac{2 x \cos (x^{2})}{\frac{1}{u} \frac{d}{d x} [\cos (x)]}$

Step 3.5.3

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

$lim_{x \to 0} \frac{2 x \cos (x^{2})}{\frac{1}{\cos (x)} \frac{d}{d x} [\cos (x)]}$

Step 3.6

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

$lim_{x \to 0} \frac{2 x \cos (x^{2})}{\frac{1}{\cos (x)} (- \sin (x))}$

Step 3.7

Combine $\frac{1}{\cos (x)}$ and $\sin (x)$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

Combine factors.

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Step 5.1

Multiply $- 1$ by $2$ .

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

Step 5.2

Combine $- 2$ and $\frac{\cos (x)}{\sin (x)}$ .

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

Step 5.3

Combine $\frac{- 2 \cos (x)}{\sin (x)}$ and $x$ .

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

Step 5.4

Combine $\frac{- 2 \cos (x) x}{\sin (x)}$ and $\cos (x^{2})$ .

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

Step 6

Convert from $\frac{- 2 \cos (x) x \cos (x^{2})}{\sin (x)}$ to $- 2 \cos (x) x \cot (x)$ .

$lim_{x \to 0} - 2 \cos (x) x \cot (x)$

Step 7

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

$- 2 lim_{x \to 0} \cos (x) x \cot (x)$

Step 8

Consider the left sided limit.

$lim_{x \to 0^{-}} \cos (x) x \cot (x)$

Step 9

Make a table to show the behavior of the function $\cos (x) x \cot (x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \cos (x) x \cot (x) \\ - 0.1 & 0.99168527 \\ - 0.01 & 0.99991666 \\ - 0.001 & 0.99999916 \end{array}$

Step 10

As the $x$ values approach $0$ , the function values approach $1$ . Thus, the limit of $\cos (x) x \cot (x)$ as $x$ approaches $0$ from the left is $1$ .

$1$

Step 11

Consider the right sided limit.

$lim_{x \to 0^{+}} \cos (x) x \cot (x)$

Step 12

Make a table to show the behavior of the function $\cos (x) x \cot (x)$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \cos (x) x \cot (x) \\ 0.1 & 0.99168527 \\ 0.01 & 0.99991666 \\ 0.001 & 0.99999916 \end{array}$

Step 13

As the $x$ values approach $0$ , the function values approach $1$ . Thus, the limit of $\cos (x) x \cot (x)$ as $x$ approaches $0$ from the right is $1$ .

$- 2 \cdot 1$

Step 14

Multiply $- 2$ by $1$ .

$- 2$