Calculus Examples

lim x \to 0 \frac{4}{1 + e^{\frac{1}{x}}}

$lim_{x \to 0} \frac{4}{1 + e^{\frac{1}{x}}}$

Step 1

Consider the left sided limit.

lim x \to 0^{-} \frac{4}{1 + e^{\frac{1}{x}}}

$lim_{x \to 0^{-}} \frac{4}{1 + e^{\frac{1}{x}}}$

Step 2

As the

x

$x$ values approach

0

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

\infty

$\infty$

Step 3

Consider the right sided limit.

lim x \to 0^{+} \frac{4}{1 + e^{\frac{1}{x}}}

$lim_{x \to 0^{+}} \frac{4}{1 + e^{\frac{1}{x}}}$

Step 4

Make a table to show the behavior of the function

\frac{4}{1 + e^{\frac{1}{x}}}

$\frac{4}{1 + e^{\frac{1}{x}}}$ as

x

$x$ approaches

0

$0$ from the right.

\begin{matrix} x & \frac{4}{1 + e^{\frac{1}{x}}} 0.1 & 0.00018159 0.01 & 0 0.001 & 0 \end{matrix}

$\begin{array}{c} x & \frac{4}{1 + e^{\frac{1}{x}}} \\ 0.1 & 0.00018159 \\ 0.01 & 0 \\ 0.001 & 0 \end{array}$

Step 5

As the

x

$x$ values approach

0

$0$ , the function values approach

0

$0$ . Thus, the limit of

\frac{4}{1 + e^{\frac{1}{x}}}

$\frac{4}{1 + e^{\frac{1}{x}}}$ as

x

$x$ approaches

0

$0$ from the right is

0

$0$ .

0

$0$

Step 6

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

Does not exist

$Does not exist$