Question

How do you find the limit at a hole?

The function I have is `lim_(x rarr 9) (x-9)/(sqrt(x+7)-4.)` How do I find the limit, if the function gives `0/0?`

Answer 1

`lim_(x->9) (x-9)/(sqrt(x+7)-4) = 8`

Explanation:

We can find this limit algebraically by eliminating a common factor that is causing the hole. It is slightly cleaner to see if we put `t = x+7` as follows...

`lim_(x->9) (x-9)/(sqrt(x+7)-4) = lim_(t->16) (t-16)/(sqrt(t)-4)`

`color(white)(lim_(x->9) (x-9)/(sqrt(x+7)-4)) = lim_(t->16) (color(red)(cancel(color(black)((sqrt(t)-4))))(sqrt(t)+4))/color(red)(cancel(color(black)((sqrt(t)-4))))`

`color(white)(lim_(x->9) (x-9)/(sqrt(x+7)-4)) = lim_(t->16) sqrt(t)+4`

`color(white)(lim_(x->9) (x-9)/(sqrt(x+7)-4)) = sqrt(16)+4`

`color(white)(lim_(x->9) (x-9)/(sqrt(x+7)-4)) = 8`