Question

How do you find the limit of `((sqrt(169-x^2))+12)/(x-5)` as x approaches 5?

Answer 1

`lim_{xto5}(sqrt(169-x^2)+12)/(x-5) = ∞`

Explanation:

Let's try direct substitution.

`(sqrt(169-x^2) + 12)/(x-5) = (sqrt(169-5^2)+ 12)/(5-5) = (sqrt(169-25)+12)/0 = (sqrt144+12)/0 = (12+12)/0 = 24/0`

So there is a "hole" at x = 5.

`lim_{xto5}(sqrt(169-x^2)+12)/(x-5) = lim_{xto5}24/0 = ∞`