Limits, Asymptotes, and End Behavior — Quiz

Question 10 • Precalculus

Which statement correctly demonstrates using limits to determine a vertical asymptote of g of x is equal to the fraction with numerator 2 times open paren x plus 4 close paren squared and denominator x squared minus 16 ?

Answer

There is a vertical asymptote at x is equal to negative 4 because limit x to 4 raised to the negative power of g times x is equal to negative infinity and limit x to 4 raised to the plus power of g times x is equal to negative infinity .

There is a vertical asymptote at x is equal to negative 4 because limit x to 4 raised to the negative power of g times x is equal to infinity and limit x to 4 raised to the plus power of g times x is equal to negative infinity .

There is a vertical asymptote at x is equal to 4 because limit x to 4 raised to the negative power of g times x is equal to negative infinity and limit x to 4 raised to the plus power of g times x is equal to infinity .

There is a vertical asymptote at x is equal to 4 because limit x to 4 raised to the negative power of g times x is equal to infinity and limit x to 4 raised to the plus power of g times x is equal to negative infinity .

Explanation

Factor the denominator: x² − 16 = (x − 4)(x + 4), so for x ≠ −4 the function simplifies to 2(x + 4)/(x − 4). Near x = 4, the numerator is positive, while the denominator approaches 0 from below on the left and from above on the right. Thus the left-hand limit is negative infinity and the right-hand limit is positive infinity, so x = 4 is a vertical asymptote. This matches choice C.

AI-written and checked against the verified answer.

Back to All Questions

More from this Course

Quiz

Quiz

Quiz