Question

graph of h
the graph of the function h is shown above. what is \\(\lim_{x \to 4} h(x)\\)?
\\(\boldsymbol{\text{a}}\\) \\(-1\\)
\\(\boldsymbol{\text{b}}\\) \\(2\\)
\\(\boldsymbol{\text{c}}\\) \\(5\\)
\\(\boldsymbol{\text{d}}\\) nonexistent

Explanation:

Step1: Recall limit definition

To find $\lim_{x \to 4} h(x)$, we check the left - hand limit and the right - hand limit as $x$ approaches 4. The limit exists if $\lim_{x \to 4^{-}} h(x)=\lim_{x \to 4^{+}} h(x)$.

Step2: Analyze left - hand limit

As $x$ approaches 4 from the left (values of $x$ less than 4), we look at the linear part of the graph. From the graph, when $x$ approaches 4 from the left, the $y$ - value (the value of the function $h(x)$) approaches 5. So, $\lim_{x \to 4^{-}} h(x) = 5$.

Step3: Analyze right - hand limit

As $x$ approaches 4 from the right (values of $x$ greater than 4), we look at the parabolic part of the graph. From the graph, when $x$ approaches 4 from the right, the $y$ - value (the value of the function $h(x)$) also approaches 5. So, $\lim_{x \to 4^{+}} h(x)=5$.

Step4: Determine the limit

Since $\lim_{x \to 4^{-}} h(x)=\lim_{x \to 4^{+}} h(x) = 5$, by the definition of the limit, $\lim_{x \to 4} h(x)=5$.

Answer:

C. 5