Example 4: Find .
If you plug in some very small values for x, you’ll see that this function approaches ∞. And it doesn’t
matter whether x is positive or negative, you still get ∞. Look at the graph of y = :
On either side of x = 0 (the y-axis), the curve approaches ∞.
Example 5: Find .
Here you have a problem. If you plug in some very small positive values for x (0.1, 0.01, 0.001, and so
on), you approach ∞. In other words, = ∞. But, if you plug in some very small negative values for
x (−0.1, −0.01, −0.001, and so on), you approach −∞. That is, = −∞. Because the right-hand limit
is not equal to the left-hand limit, the limit does not exist.
Look at the graph of .