Question

Let f be a function such that f (x) = g(x) , where h and g are continuous for all real numbers. h(x) Which one of the following statements must be true? (A) f must be a continuous function. (B) If h(c) = 0 , then x = c must be a vertical asymptote. (C)If limf(x)=¥then x=cisaverticalasymptote. ìx2-2x+3, x£1 î x®c (D) If g(c)=0and h(c)=0then limf(x)doesnotexist.

Answer 1

The correct statement is:

(D) If g(c) = 0 and h(c) = 0, then lim f(x) does not exist.

This statement makes sense in the context of continuous functions. If both g(x) and h(x) are continuous functions for all real numbers and you have specific points where they are both zero (g(c) = 0 and h(c) = 0), this doesn't necessarily mean that their ratio f(x) = g(x) / h(x) will have a limit as x approaches c. In this case, the limit of f(x) may not exist due to the indeterminate form 0/0, which can occur at points where both g(x) and h(x) are zero but cancel each other out differently as x approaches c.

Option (A) is not necessarily true, as the continuity of g(x) and h(x) does not guarantee the continuity of their ratio f(x).

Option (B) and (C) involve vertical asymptotes but are not necessarily true based on the information given in the question. The existence of vertical asymptotes depends on the behavior of the functions g(x) and h(x) around c and cannot be determined solely from the information provided.

So, option (D) is the most accurate statement based on the given information.