Calculus Examples

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

Step 1

Set up the limit as a left-sided limit.

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

Step 2

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

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

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

Step 4

Evaluate the right-sided limit.

Tap for more steps...

Step 4.1

Apply L'Hospital's rule.

Tap for more steps...

Step 4.1.1

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

Tap for more steps...

Step 4.1.1.1

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

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

Step 4.1.1.2

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

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

Step 4.1.1.3

As the $x$ values approach $0$ from the right, the function values increase 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.

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

Step 4.1.3

Find the derivative of the numerator and denominator.

Tap for more steps...

Step 4.1.3.1

Differentiate the numerator and denominator.

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

Step 4.1.3.2

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

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

Step 4.1.3.3

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

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

Step 4.1.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.1.5

Multiply $\frac{1}{x}$ by $\frac{1}{- \csc^{2} (x)}$ .

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

Step 4.1.6

Cancel the common factor of $1$ and $- 1$ .

Tap for more steps...

Step 4.1.6.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 4.1.6.2

Move the negative in front of the fraction.

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

Step 4.2

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

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

Step 4.3

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

$- 0$

Step 4.4

Multiply $- 1$ by $0$ .

$0$

Step 5

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

$Does not exist$