Question

Find the limits. \(f(x)=x+7, g(x)=x^{2}\) (a) \(\lim _{x \rightarrow-3} f(x)\) (b) \(\lim _{x \rightarrow 4} g(x)\) (c) \(\lim _{x \rightarrow-3} g(f(x))\)

Short answer

\(\lim_{x \rightarrow -3} f(x) = 4\), \(\lim_{x \rightarrow 4} g(x) = 16\), and \(\lim_{x \rightarrow -3} g(f(x)) = 16\)

Step-by-step solution

Identify the limit being evaluated for each function

For each limit in the exercise, we need to identify which function is being evaluated and what value x is approaching. (a) \(f(x)=x+7\) as \(x \rightarrow -3\), (b) \(g(x)=x^{2}\) as \(x \rightarrow 4\), and (c) \(g(f(x))=g(x+7)\) as \(x \rightarrow -3\).

Evaluate the limit for the linear function f(x)

To find the limit of \(f(x)=x+7\) as \(x \rightarrow -3\), substitute -3 into the function to get \(f(-3)=-3+7=4\). So, \(\lim_{x \rightarrow -3} f(x) = 4\)

Evaluate the limit for the quadratic function g(x)

To find the limit of \(g(x)=x^{2}\) as \(x \rightarrow 4\) substitute 4 into the function to get \(g(4)=4^{2}=16\). So, \(\lim_{x \rightarrow 4} g(x) = 16\)

Evaluate the limit for the composition of functions

To find the limit of \(g(f(x))=g(x+7)\) as \(x \rightarrow -3\), substitute -3 into the inside function f(x) to get \(f(-3) = -3+7 = 4\). Then substitute 4 into \(g(x)\) to get \(g(4)=4^{2}=16\). So, \(\lim_{x \rightarrow -3} g(f(x)) = 16\)