Calculus Examples

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

Step 1

Consider the left sided limit.

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

Step 2

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

$\begin{array}{c} x & e^{\frac{1}{x}} \\ - 0.1 & 0.00004539 \\ - 0.01 & 0 \\ - 0.001 & 0 \end{array}$

Step 3

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

$0$

Step 4

Consider the right sided limit.

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

Step 5

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

$\infty$

Step 6

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

$Does not exist$