Question

suppose f and g are continuous functions such that g(2) = 5 and lim_{x→2}3f(x)+f(x)g(x) = 24. find f(2).

Explanation:

Step1: Use continuity property

Since \(f\) and \(g\) are continuous functions, \(\lim_{x
ightarrow 2}[3f(x)+f(x)g(x)] = 3f(2)+f(2)g(2)\).

Step2: Substitute known values

We know \(g(2) = 5\) and \(\lim_{x
ightarrow 2}[3f(x)+f(x)g(x)]=24\). So \(3f(2)+f(2)\times5 = 24\).

Step3: Combine like - terms

Factor out \(f(2)\) on the left - hand side: \(f(2)(3 + 5)=24\), which simplifies to \(8f(2)=24\).

Step4: Solve for \(f(2)\)

Divide both sides by 8: \(f(2)=\frac{24}{8}=3\).

Answer:

3