Question

I’m not sure how to do this

Answer 1

Answer: lim as x → -5 of f(x) and g(x) = 300

Domain of f(x) and g(x) is All Real Numbers

f(-5) = 200

g(-5) = 300

Explanation:

The limit of f(x) as x approaches -5:

`f(x) =(4x^3+500)/(x+5)\\\\\\.\qquad =(4(x^3+125))/(x+5)\\\\\\.\qquad =(4(x^3+5^3))/(x+5)\qquad \text{we can factor the cubic}\\\\\\.\qquad =(4(x+5)(x^2-5x+25))/(x+5)\\\\\\.\qquad =4(x^2-5x+25)\\\\\\\text{as x approaches -5, f(x) = }4[(-5)^2-5(-5)+25]\\\\.\qquad \qquad \qquad \qquad \qquad =4(25 + 25 + 25)\\\\.\qquad \qquad \qquad \qquad \qquad =4(75)\\\\.\qquad \qquad \qquad \qquad \qquad =300`

The limit of g(x) as x approaches -5:

g(x) = 4x² - 20x + 100

= 4(-5)² - 20(-5) + 100

= 100 + 100 + 100

= 300

There is a restriction at x = -5 for f(x), however, that discontinuity has been filled with the 200 at x = -5. So the domain is ALL REAL NUMBERs.

There are no restrictions on x for g(x) so the domain is ALL REAL NUMBERs.