Question

If `lim_(xrarr2)(f(x)-4)/(x-2)=C` for constant C. Find `lim_(xrarr2)`f(x). What theorems were used to come to conclusion?

Answer 1

4

Explanation:

We can use the properties

`lim_{x to a}[f(x)+g(x)] = lim_{x to a} f(x) +lim_{x to a}g(x)`

and

`lim_{x to a}[f(x)g(x)] = lim_{x to a} f(x) lim_{x to a}g(x)`

provided both the limits on the right exists.

Thus

`lim_{x to 2}(f(x)-4) = lim_{x to 2} (f(x)-4)/(x-2) lim_{x to 2}(x-2)`
`qquad qquad qquad qquad = C times 0 = 0`

Then

`lim_{x to 2}f(x) = lim_{x to 2}(f(x)-4)+lim_{x to 2}4`
`qquad = 0+4 = 4`