Calculus Examples

$lim_{x \to 0} {(1 + x)}^{\cot (x)}$

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite ${(1 + x)}^{\cot (x)}$ as $e^{\ln ({(1 + x)}^{\cot (x)})}$ .

$lim_{x \to 0} e^{\ln ({(1 + x)}^{\cot (x)})}$

Step 1.2

Expand $\ln ({(1 + x)}^{\cot (x)})$ by moving $\cot (x)$ outside the logarithm.

$lim_{x \to 0} e^{\cot (x) \ln (1 + x)}$

Step 2

Move the limit into the exponent.

$e^{lim_{x \to 0} \cot (x) \ln (1 + x)}$

Step 3

Consider the left sided limit.

$lim_{x \to 0^{-}} \cot (x) \ln (1 + x)$

Step 4

Make a table to show the behavior of the function $\cot (x) \ln (1 + x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \cot (x) \ln (1 + x) \\ - 0.1 & 1.05009079 \\ - 0.01 & 1.00500008 \\ - 0.001 & 1.0005 \\ - 0.0001 & 1.00005 \\ - 0.00001 & 1.00000499 \end{array}$

Step 5

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

$1$

Step 6

Consider the right sided limit.

$lim_{x \to 0^{+}} \cot (x) \ln (1 + x)$

Step 7

Make a table to show the behavior of the function $\cot (x) \ln (1 + x)$ as $x$ approaches $0$ from the right.

$\begin{array}{c} x & \cot (x) \ln (1 + x) \\ 0.1 & 0.94992267 \\ 0.01 & 0.99499991 \\ 0.001 & 0.99949999 \\ 0.0001 & 0.99995 \\ 0.00001 & 0.99999499 \end{array}$

Step 8

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

$e^{1}$

Step 9

Simplify.

$e$

Step 10

The result can be shown in multiple forms.

Exact Form:

$e$

Decimal Form:

$2.71828182 \dots$