Calculus Examples

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

Step 1

Consider the left sided limit.

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

Step 2

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

$\infty$

Step 3

Consider the right sided limit.

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

Step 4

Make a table to show the behavior of the function $\frac{4}{1 + e^{\frac{1}{x}}}$ as $x$ approaches $0$ from the right.

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

Step 5

As the $x$ values approach $0$ , the function values approach $0$ . Thus, the limit of $\frac{4}{1 + e^{\frac{1}{x}}}$ as $x$ approaches $0$ from the right is $0$ .

$0$

Step 6

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

$Does not exist$