Question

ii. use the graph to answer the limit statements.
graph of h(x)
a) \\(\lim\limits_{x\to 2^-} h(x) =\\)
b) \\(\lim\limits_{x\to 2^+} h(x) =\\)
c) \\(\lim\limits_{x\to 3^-} h(x) =\\)
d) \\(\lim\limits_{x\to 3^+} h(x) =\\)

Explanation:

Part (a): $\boldsymbol{\lim_{x \to 2^-} h(x)}$

Step1: Analyze left - hand limit at \(x = 2\)

To find the left - hand limit as \(x\) approaches \(2\) (denoted as \(x\to2^-\)), we look at the behavior of the function \(h(x)\) for values of \(x\) that are less than \(2\) and getting closer to \(2\). From the graph, as \(x\) approaches \(2\) from the left (values like \(1.9,1.99,1.999,\cdots\)), the function \(h(x)\) is approaching \(- 4\).

Step1: Analyze right - hand limit at \(x = 2\)

To find the right - hand limit as \(x\) approaches \(2\) (denoted as \(x\to2^+\)), we look at the behavior of the function \(h(x)\) for values of \(x\) that are greater than \(2\) and getting closer to \(2\). From the graph, as \(x\) approaches \(2\) from the right (values like \(2.1,2.01,2.001,\cdots\)), the function \(h(x)\) is approaching \(7\).

Step1: Analyze left - hand limit at \(x = 3\)

To find the left - hand limit as \(x\) approaches \(3\) (denoted as \(x\to3^-\)), we look at the behavior of the function \(h(x)\) for values of \(x\) that are less than \(3\) and getting closer to \(3\). From the graph, as \(x\) approaches \(3\) from the left (values like \(2.9,2.99,2.999,\cdots\)), the function \(h(x)\) is approaching \(3\) (we can see the open circle at \(x = 3\) from the left - hand side, and the function values are approaching \(3\)).

Answer:

\(\lim_{x\to2^-}h(x)=- 4\)

Part (b): $\boldsymbol{\lim_{x \to 2^+} h(x)}$