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Chapter 1: Problem 18

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

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

Step by step solution

01

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 \(x \rightarrow -3\).
02

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, \(\lim_{x \rightarrow -3} f(x) = 4\)
03

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, \(\lim_{x \rightarrow 4} g(x) = 16\)
04

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

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