Calculus Examples

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

Step 1

Set up the limit as a left-sided limit.

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

Step 2

Evaluate the limits by plugging in the value for the variable.

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Step 2.1

Evaluate the limit of $\frac{(\cot (x))}{\ln (x)}$ by plugging in $0$ for $x$ .

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

Step 2.2

Rewrite $\cot (0)$ in terms of sines and cosines.

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

Step 2.3

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

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

Step 2.4

Since $\frac{\frac{\cos (0)}{0}}{\ln (0)}$ is undefined, the limit does not exist.

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

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

Step 4

Evaluate the right-sided limit.

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Step 4.1

Apply L'Hospital's rule.

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Step 4.1.1

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

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Step 4.1.1.1

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

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

Step 4.1.1.2

As the $x$ values approach $0$ from the right, the function values increase without bound.

$\frac{\infty}{lim_{x \to 0^{+}} \ln (x)}$

Step 4.1.1.3

As $x$ approaches $0$ from the right side, $\ln (x)$ decreases without bound.

$\frac{\infty}{- \infty}$

Step 4.1.1.4

Infinity divided by infinity is undefined.

Undefined

$\frac{\infty}{- \infty}$

Step 4.1.2

Since $\frac{\infty}{- \infty}$ 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.

$lim_{x \to 0^{+}} \frac{(\cot (x))}{\ln (x)} = lim_{x \to 0^{+}} \frac{\frac{d}{d x} [\cot (x)]}{\frac{d}{d x} [\ln (x)]}$

Step 4.1.3

Find the derivative of the numerator and denominator.

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Step 4.1.3.1

Differentiate the numerator and denominator.

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

Step 4.1.3.2

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

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

Step 4.1.3.3

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

$lim_{x \to 0^{+}} \frac{- \csc^{2} (x)}{\frac{1}{x}}$

Step 4.1.4

Multiply the numerator by the reciprocal of the denominator.

$lim_{x \to 0^{+}} - \csc^{2} (x) x$

Step 4.2

Rewrite $- \csc^{2} (x) x$ as $\frac{- x}{\csc^{- 2} (x)}$ .

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

Step 4.3

Apply L'Hospital's rule.

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Step 4.3.1

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

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Step 4.3.1.1

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

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

Step 4.3.1.2

Evaluate the limit of the numerator.

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Step 4.3.1.2.1

Move the term $- 1$ outside of the limit because it is constant with respect to $x$ .

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

Step 4.3.1.2.2

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

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

Step 4.3.1.3

Evaluate the limit of the denominator.

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Step 4.3.1.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{0}{lim_{x \to 0^{+}} \frac{1}{\csc^{2} (x)}}$

Step 4.3.1.3.2

Since its numerator approaches a real number while its denominator is unbounded, the fraction $\frac{1}{\csc^{2} (x)}$ approaches $0$ .

$\frac{0}{0}$

Step 4.3.1.3.3

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

Undefined

$\frac{0}{0}$

Step 4.3.1.4

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

Undefined

$\frac{0}{0}$

Step 4.3.2

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.

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

Step 4.3.3

Find the derivative of the numerator and denominator.

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Step 4.3.3.1

Differentiate the numerator and denominator.

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

Step 4.3.3.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 4.3.3.3

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

$lim_{x \to 0^{+}} \frac{- 1 \cdot 1}{\frac{d}{d x} [\csc^{- 2} (x)]}$

Step 4.3.3.4

Multiply $- 1$ by $1$ .

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

Step 4.3.3.5

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

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Step 4.3.3.5.1

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

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

Step 4.3.3.5.2

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

$lim_{x \to 0^{+}} \frac{- 1}{- 2 u^{- 3} \frac{d}{d x} [\csc (x)]}$

Step 4.3.3.5.3

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

$lim_{x \to 0^{+}} \frac{- 1}{- 2 \csc^{- 3} (x) \frac{d}{d x} [\csc (x)]}$

Step 4.3.3.6

The derivative of $\csc (x)$ with respect to $x$ is $- \csc (x) \cot (x)$ .

$lim_{x \to 0^{+}} \frac{- 1}{- 2 \csc^{- 3} (x) (- \csc (x) \cot (x))}$

Step 4.3.3.7

Multiply $- 1$ by $- 2$ .

$lim_{x \to 0^{+}} \frac{- 1}{2 \csc^{- 3} (x) (\csc (x) \cot (x))}$

Step 4.3.3.8

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

$lim_{x \to 0^{+}} \frac{- 1}{2 (\csc^{1} (x) \csc^{- 3} (x)) \cot (x)}$

Step 4.3.3.9

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

$lim_{x \to 0^{+}} \frac{- 1}{2 {\csc (x)}^{1 - 3} \cot (x)}$

Step 4.3.3.10

Subtract $3$ from $1$ .

$lim_{x \to 0^{+}} \frac{- 1}{2 \csc^{- 2} (x) \cot (x)}$

Step 4.3.3.11

Simplify.

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Step 4.3.3.11.1

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

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

Step 4.3.3.11.2

Change the sign of the exponent by rewriting the base as its reciprocal.

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

Step 4.3.3.11.3

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

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

Step 4.3.3.11.4

Cancel the common factor of $\sin (x)$ .

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Step 4.3.3.11.4.1

Factor $\sin (x)$ out of $2 \sin^{2} (x)$ .

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

Step 4.3.3.11.4.2

Cancel the common factor.

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

Step 4.3.3.11.4.3

Rewrite the expression.

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

Step 4.3.3.11.5

Apply the sine double-angle identity.

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

Step 4.3.4

Separate fractions.

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

Step 4.3.5

Convert from $\frac{1}{\sin (2 x)}$ to $\csc (2 x)$ .

$lim_{x \to 0^{+}} \frac{- 1}{1} \csc (2 x)$

Step 4.3.6

Divide $- 1$ by $1$ .

$lim_{x \to 0^{+}} - \csc (2 x)$

Step 4.4

Since the function $\csc (2 x)$ approaches $\infty$ , the negative constant $- 1$ times the function approaches $- \infty$ .

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Step 4.4.1

Consider the limit with the constant multiple $- 1$ removed.

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

Step 4.4.2

As the $x$ values approach $0$ from the right, the function values increase without bound.

$\infty$

Step 4.4.3

Since the function $\csc (2 x)$ approaches $\infty$ , the negative constant $- 1$ times the function approaches $- \infty$ .

$- \infty$

Step 5

If either of the one-sided limits does not exist, the limit does not exist.

$Does not exist$