Answer :
To determine which of the given statements must be true, given that [tex]\( f \)[/tex] is a continuous function and [tex]\( f(3) = 7 \)[/tex], let's analyze each option one by one:
(A) [tex]\( \lim _{x \rightarrow 3} f(3x) = 9 \)[/tex]
- When [tex]\( x \rightarrow 3 \)[/tex], [tex]\( 3x \rightarrow 9 \)[/tex]. Thus, [tex]\( \lim _{x \rightarrow 3} f(3x) = f(9) \)[/tex]. However, we do not have any information about [tex]\( f(9) \)[/tex]. Therefore, this statement cannot be determined to be true.
(B) [tex]\( \lim _{x \rightarrow 3} f(2x) = 14 \)[/tex]
- When [tex]\( x \rightarrow 3 \)[/tex], [tex]\( 2x \rightarrow 6 \)[/tex]. Thus, [tex]\( \lim _{x \rightarrow 3} f(2x) = f(6) \)[/tex]. We do not know the value of [tex]\( f(6) \)[/tex]. Therefore, this statement cannot be determined to be true.
(C) [tex]\( \lim _{x \rightarrow 3} \frac{f(x) - f(3)}{x-3} = 7 \)[/tex]
- This statement represents the derivative of [tex]\( f \)[/tex] at [tex]\( x = 3 \)[/tex]. Written in the form [tex]\( f'(3) = 7 \)[/tex], we would need additional information about the function [tex]\( f \)[/tex] to confirm this. Therefore, this statement cannot be determined to be true from the given information.
(D) [tex]\( \lim _{x \rightarrow 3} f(x^2) = 49 \)[/tex]
- When [tex]\( x \rightarrow 3 \)[/tex], [tex]\( x^2 \rightarrow 9 \)[/tex]. Thus, [tex]\( \lim _{x \rightarrow 3} f(x^2) = f(9) \)[/tex]. Since we do not know [tex]\( f(9) \)[/tex], this statement cannot be determined to be true.
(E) [tex]\( \lim _{x \rightarrow 3} (f(x))^2 = 49 \)[/tex]
- When [tex]\( x \rightarrow 3 \)[/tex], [tex]\( f(x) \rightarrow f(3) \)[/tex]. We know [tex]\( f(3) = 7 \)[/tex], so [tex]\( (f(x))^2 \rightarrow (f(3))^2 = 7^2 = 49 \)[/tex]. Thus, this statement is true.
Therefore, the correct answer is (E) [tex]\( \lim _{x \rightarrow 3} (f(x))^2 = 49 \)[/tex].
(A) [tex]\( \lim _{x \rightarrow 3} f(3x) = 9 \)[/tex]
- When [tex]\( x \rightarrow 3 \)[/tex], [tex]\( 3x \rightarrow 9 \)[/tex]. Thus, [tex]\( \lim _{x \rightarrow 3} f(3x) = f(9) \)[/tex]. However, we do not have any information about [tex]\( f(9) \)[/tex]. Therefore, this statement cannot be determined to be true.
(B) [tex]\( \lim _{x \rightarrow 3} f(2x) = 14 \)[/tex]
- When [tex]\( x \rightarrow 3 \)[/tex], [tex]\( 2x \rightarrow 6 \)[/tex]. Thus, [tex]\( \lim _{x \rightarrow 3} f(2x) = f(6) \)[/tex]. We do not know the value of [tex]\( f(6) \)[/tex]. Therefore, this statement cannot be determined to be true.
(C) [tex]\( \lim _{x \rightarrow 3} \frac{f(x) - f(3)}{x-3} = 7 \)[/tex]
- This statement represents the derivative of [tex]\( f \)[/tex] at [tex]\( x = 3 \)[/tex]. Written in the form [tex]\( f'(3) = 7 \)[/tex], we would need additional information about the function [tex]\( f \)[/tex] to confirm this. Therefore, this statement cannot be determined to be true from the given information.
(D) [tex]\( \lim _{x \rightarrow 3} f(x^2) = 49 \)[/tex]
- When [tex]\( x \rightarrow 3 \)[/tex], [tex]\( x^2 \rightarrow 9 \)[/tex]. Thus, [tex]\( \lim _{x \rightarrow 3} f(x^2) = f(9) \)[/tex]. Since we do not know [tex]\( f(9) \)[/tex], this statement cannot be determined to be true.
(E) [tex]\( \lim _{x \rightarrow 3} (f(x))^2 = 49 \)[/tex]
- When [tex]\( x \rightarrow 3 \)[/tex], [tex]\( f(x) \rightarrow f(3) \)[/tex]. We know [tex]\( f(3) = 7 \)[/tex], so [tex]\( (f(x))^2 \rightarrow (f(3))^2 = 7^2 = 49 \)[/tex]. Thus, this statement is true.
Therefore, the correct answer is (E) [tex]\( \lim _{x \rightarrow 3} (f(x))^2 = 49 \)[/tex].
