Calculus Examples

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

Step 1

Set up the limit as a left-sided limit.

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

Step 2

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

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Step 2.1

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

$\ln (0) \cot (0)$

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

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

Step 4

Evaluate the right-sided limit.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

$- \infty \cdot \infty$

Step 4.4

Infinity times infinity is infinity.

$- \infty$

Step 5

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

$Does not exist$