Steps on finding out which statement for the continuous function is true?

Question

If f is a continuous function such that $f(2) = 6$, which of the following statements must be true?

(A) $\lim\limits_{x \to 1}f(2x) = 3$

(B) $\lim\limits_{x \to 2} f(2x) = 12$

(C) $\lim\limits_{x \to 2} \frac{f(x)-f(2)}{x-2} = 6$

(D) $\lim\limits_{x \to 2} f(x^2) = 36$

(E) $\lim\limits_{x \to 2} (f(x))^2 = 36$

Please show and explain the necessary steps to arrive at the correct answer. Additionally, if anyone could tell me under what unit or lesson in AP Calculus AB this type of question this would fall under, that would be great!

Thank you.

Answer 1

(E) is true. Here the explanation The concept is that composition of continuous functions is continuous and a function is continuous iff, $$\lim_{x\rightarrow a}f(x)=f(a)$$ So,

(A)the result of the limit is $f(2)=6$

(B)the result is $f(2\dot{}2)=f(4)$, value we don't know

(C)the result is undetermined $(0/0)$. We could calculate this limit is we knew the function is derivable (and the derivative also)

(D)the result is $f(2^2)=f(4)$, unknown

(E)the result is $f(2)^2=6^2=36$