Solve the following limit

Question

Solve the following limit: lim x→4 (x^2 − 4x/ x^ 2 − 16)

Answer 1

lim x→4 (x^2 − 4x/ x^ 2 − 16)

To start we see that if we evaluate at x = 4 we would get zero in the denominator, so that motivates us to do a little bit of algebra on this problem:

Rewrite the numerator: x^2-4x = x(x-4)

Rewrite the denominator: x^2-16 = (x-4)(x+4)

Now lim x→4(x^2-4x)/(x^2-16) =lim x→4 (x(x-4))/((x-4)(x+4)) = lim x→4 x/(x+4) because the (x-4) term cancels.

From here we can evaluate at 4, so lim x→4 x/(x+4) = 4/(4+4) = 4/8 = 1/2.

Answer 2

Factor this

[x(x-4)]/[(x-4)(x+4)] Cancel the x-4

lim x→4 x/(x+4) Plug in 4

4/(4+4) = 4/8 = 1/2