Evaluate following limits.

**Solution**

Let's begin with the right-hand limit. For this limit we have,

x > 4 ⇒ 4 - x < 0 ⇒ ( 4 - x )^{3} = 0 also, 4 - x → 0 as x → 4 . Therefore, we contain a positive constant divided by an increasingly small -ve number. The results will be an increasingly large -ve number and hence it looks like the right-hand limit will be negative infinity.

For the left-handed limit we have following,

x < 4 ⇒ 4 - x > 0 ⇒ ( 4 - x )^{3 } > 0 and still we have, 4 - x → 0 as x → 4 . In this case we contain a positive constant divided by an increasingly small +ve number. The results will be an increasingly large positive number and hence it looks like the left-hand limit will be positive infinity.

The normal limit will not present since the two one-sided limits are not the similar. The official answers to this example are then,

Following is a quick sketch to verify our limits.

**Facts**

Given the functions f ( x )& g ( x ) assume we have,

for some real numbers c & L. Then,