Calculus Examples

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

Step 1

Use the properties of logarithms to simplify the limit.

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Step 1.1

Rewrite

${(2 + x)}^{\frac{1}{x}}$ as

$e^{\ln ({(2 + x)}^{\frac{1}{x}})}$ .

$lim_{x \to 0} e^{\ln ({(2 + x)}^{\frac{1}{x}})}$

Step 1.2

Expand

$\ln ({(2 + x)}^{\frac{1}{x}})$ by moving

$\frac{1}{x}$ outside the logarithm.

$lim_{x \to 0} e^{\frac{1}{x} \ln (2 + x)}$

Step 2

Combine

$\frac{1}{x}$ and

$\ln (2 + x)$ .

$lim_{x \to 0} e^{\frac{\ln (2 + x)}{x}}$

Step 3

Consider the left sided limit.

$lim_{x \to 0^{-}} e^{\frac{\ln (2 + x)}{x}}$

Step 4

Make a table to show the behavior of the function

$e^{\frac{\ln (2 + x)}{x}}$ as

$x$ approaches

$0$ from the left.

$\begin{array}{c} x & e^{\frac{\ln (2 + x)}{x}} \\ - 0.1 & 0.00163103 \\ - 0.01 & 0 \\ - 0.001 & 0 \end{array}$

Step 5

As the

$x$ values approach

$0$ , the function values approach

$0$ . Thus, the limit of

$e^{\frac{\ln (2 + x)}{x}}$ as

$x$ approaches

$0$ from the left is

$0$ .

$0$

Step 6

Consider the right sided limit.

$lim_{x \to 0^{+}} e^{\frac{\ln (2 + x)}{x}}$

Step 7

As the

$x$ values approach

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

$\infty$

Step 8

Since the left sided and right sided limits are not equal, the limit does not exist.

$Does not exist$