Calculus Examples

$lim_{x \to 0} {(e^{x} - x)}^{\frac{1}{\tan (x)}}$

Step 1

Apply trigonometric identities.

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Step 1.1

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

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

Step 1.2

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

$lim_{x \to 0} {(e^{x} - x)}^{\frac{\cos (x)}{\sin (x)}}$

Step 1.3

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 2

Use the properties of logarithms to simplify the limit.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Move the limit into the exponent.

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

Step 4

Consider the left sided limit.

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

Step 5

Make a table to show the behavior of the function $\cot (x) \ln (e^{x} - x)$ as $x$ approaches $0$ from the left.

$\begin{array}{c} x & \cot (x) \ln (e^{x} - x) \\ - 0.1 & - 0.04809658 \\ - 0.01 & - 0.00498308 \\ - 0.001 & - 0.00049983 \\ - 0.0001 & - 0.00004999 \end{array}$

Step 6

As the $x$ values approach $0$ , the function values approach $0$ . Thus, the limit of $\cot (x) \ln (e^{x} - x)$ as $x$ approaches $0$ from the left is $0$ .

$0$

Step 7

Consider the right sided limit.

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

Step 8

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

$\begin{array}{c} x & \cot (x) \ln (e^{x} - x) \\ 0.1 & 0.05140391 \\ 0.01 & 0.00501641 \\ 0.001 & 0.00050016 \\ 0.0001 & 0.00005 \\ 0.00001 & 0.000005 \end{array}$

Step 9

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

$e^{0}$

Step 10

Anything raised to $0$ is $1$ .

$1$