Question

Suppose f and g are continuous functions such that g(7) = 2 and lim x → 7 [3f(x)f(x)g(x)] = 25. Find f(7).

Answer 1

The given limit expression can be simplified to
`\( \lim_{{x \to 7}} 3f(x)^2 g(x) = 25 \). Substituting \( g(7) = 2 \), we get \( \lim_{{x \to 7}} 3f(x)^2 \cdot 2 = 25 \). Solving for \( f(7) \), we find \( f(7) = 5 \).`

Step-by-step explanation:

We are given tha
`t \( g(7) = 2 \) and \( \lim_{{x \to 7}} [3f(x)f(x)g(x)] = 25 \).`We need to find
`\( f(7) \)`. The given limit can be expressed as
`\( \lim_{{x \to 7}} 3f(x)^2 g(x) = 25 \).`

Since
`\( \lim_{{x \to c}} g(x) = g(c) \) when \( g(x) \) is continuous at \( x = c \), we can substitute \( g(7) = 2 \) into the limit expression:\[ \lim_{{x \to 7}} 3f(x)^2 \cdot 2 = 25 \]`

Solving for
`\( \lim_{{x \to 7}} 3f(x)^2 = (25)/(2) \). Now, we know \( \lim_{{x \to c}} f(x)^2 = [f(c)]^2 \) when \( f(x) \) is continuous at \( x = c \). So,\[ 3[f(7)]^2 = (25)/(2) \]`

Solving for
`\( f(7) \), we get \( f(7) = 5 \).`

In summary, by using the properties of limits and continuity, we found that \
`( f(7) = 5 \)`is the solution to the given problem.