Calculus Examples

$lim_{x \to 3} e^{\frac{2}{3 - x}}$

Step 1

Consider the left sided limit.

$lim_{x \to 3^{-}} e^{\frac{2}{3 - x}}$

Step 2

As the $x$ values approach $3$ from the left, the function values increase without bound.

$\infty$

Step 3

Consider the right sided limit.

$lim_{x \to 3^{+}} e^{\frac{2}{3 - x}}$

Step 4

Make a table to show the behavior of the function $e^{\frac{2}{3 - x}}$ as $x$ approaches $3$ from the right.

$\begin{array}{c} x & e^{\frac{2}{3 - x}} \\ 3.1 & 0 \\ 3.01 & 0 \\ 3.001 & 0 \end{array}$

Step 5

As the $x$ values approach $3$ , the function values approach $0$ . Thus, the limit of $e^{\frac{2}{3 - x}}$ as $x$ approaches $3$ from the right is $0$ .

$0$

Step 6

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

$Does not exist$