Question

Let f(x) = (x + 7)^2 / (x^2 - 49). Find:

(a) lim(x -> -7) f(x)
(b) lim(x -> 0) f(x)
(c) lim(x -> 7) f(x)

Answer 1

To find the limits of the given function, substitute the given value of x into the function and simplify. Indeterminate forms require the use of L'Hopital's Rule to find the limit. The limits at x = -7 and x = 7 are undefined.

Step-by-step explanation:

To find the limits of the given function, we substitute the given value of x into the function and simplify.

(a) limx → -7 f(x) = (-7 + 7)2 / ((-7)2 - 49) = 0 / 0

This is an indeterminate form, and we can use L'Hopital's Rule to find the limit.

By differentiating the numerator and denominator and taking the limit again, we get limx → -7 f(x) = 2 / (-7 + 7) = 2 / 0, which is undefined.

(b) limx → 0 f(x) = (0 + 7)2 / (02 - 49) = 49 / (-49) = -1

(c) limx → 7 f(x) = (7 + 7)2 / (72 - 49) = 196 / 0

This is also an indeterminate form, so we use L'Hopital's Rule again. By differentiating the numerator and denominator and taking the limit, we get limx → 7 f(x) = 2 / (7 - 7) = 2 / 0, which is undefined.