Question

Find the indicated limit, if it exists. lim x-->0 f(x), f(x)=(9-x^2, x<0, 9, x=0, -4x+9, x>0)

Answer 1

Choices are The limit does not exist, 9, -4, -13

Answer 2

Well, we can see that the function value at 0 is equal to 9, but that doesn't mean the limit is 9. We want to show that the limit as x approaches zero from the left is equal to the limit as we approach zero from the right.

Since f is definited to be 9-x^2 for x < 0, we follow the path along f = 9-x^2 as we approach from the left. So taking that limit:
\[\lim_{x \rightarrow 0^{-}}(9-x^{2}) = 9-0 = 9\]
If we approach from the right, then we're approaching along the path f = -4x + 9. So th limit from the right uses this function and we have:
\[\lim_{x \rightarrow 0^{+}}(-4x+9) = 0 + 9 = 9\]
Thus the limit from the left is equal to the limit from the right and that value is equal to 9.