Calculus Examples

$lim_{x \to - 3} \frac{- 4 | x + 3 |}{- 3 (x + 3)}$

Step 1

Consider the left sided limit.

$lim_{x \to {(- 3)}^{-}} \frac{- 4 | x + 3 |}{- 3 (x + 3)}$

Step 2

Make a table to show the behavior of the function $\frac{- 4 | x + 3 |}{- 3 (x + 3)}$ as $x$ approaches $- 3$ from the left.

$\begin{array}{c} x & \frac{- 4 | x + 3 |}{- 3 (x + 3)} \\ - 3.1 & - 1.\overline{3} \\ - 3.01 & - 1.\overline{3} \\ - 3.001 & - 1.\overline{3} \end{array}$

Step 3

As the $x$ values approach $- 3$ , the function values approach $- 1.333$ . Thus, the limit of $\frac{- 4 | x + 3 |}{- 3 (x + 3)}$ as $x$ approaches $- 3$ from the left is $- 1.333$ .

$- 1.333$

Step 4

Consider the right sided limit.

$lim_{x \to {(- 3)}^{+}} \frac{- 4 | x + 3 |}{- 3 (x + 3)}$

Step 5

Make a table to show the behavior of the function $\frac{- 4 | x + 3 |}{- 3 (x + 3)}$ as $x$ approaches $- 3$ from the right.

$\begin{array}{c} x & \frac{- 4 | x + 3 |}{- 3 (x + 3)} \\ - 2.9 & 1.\overline{3} \\ - 2.99 & 1.\overline{3} \\ - 2.999 & 1.\overline{3} \end{array}$

Step 6

As the $x$ values approach $- 3$ , the function values approach $1.333$ . Thus, the limit of $\frac{- 4 | x + 3 |}{- 3 (x + 3)}$ as $x$ approaches $- 3$ from the right is $1.333$ .

$1.333$

Step 7

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

$Does not exist$