Calculus Examples

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

Step 1

Set up the limit as a left-sided limit.

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

Step 2

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

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Step 2.1

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

$\cot (\ln (0))$

Step 2.2

Since $\cot (\ln (0))$ is undefined, the limit does not exist.

$Does not exist$

Step 3

Set up the limit as a right-sided limit.

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

Step 4

Evaluate the right-sided limit.

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Step 4.1

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

$\begin{array}{c} x & \cot (\ln (x)) \\ 0.1 & 0.89814404 \\ 0.01 & - 0.10763155 \\ 0.001 & - 1.38728817 \\ 0.0001 & 4.59166223 \\ 0.00001 & 0.56904996 \\ 0.000001 & - 0.3332287 \\ 0.0000001 & - 2.29996828 \\ 0.00000001 & 2.18693809 \\ 0 & 0.31253152 \\ 0 & - 0.59413247 \end{array}$

Step 4.2

As the $x$ values approach $0$ , the function values approach $- 0.594$ . Thus, the limit of $\cot (\ln (x))$ as $x$ approaches $0$ from the right is $- 0.594$ .

$- 0.594$

Step 5

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

$Does not exist$