Question

Use a graphing utility to graph the function and visually estimate the limits. \(h(x)=x^{2}-5 x\) (a) \(\lim _{x \rightarrow 5} h(x)\) (b) \(\lim _{x \rightarrow-1} h(x)\)

Short answer

The exact limits can't be given as we are visually estimating from the graph. However, the limit of \(h(x)=x^{2}-5 x\) as \(x\) approaches 5 and -1 will be the value the function \(h(x)\) seems to be heading towards at \(x=5\) and \(x=-1\) respectively, from the visual analysis of the graph.

Step-by-step solution

Graphing the function

Use a graphing utility to plot the function \(h(x)=x^{2}-5 x\). This will give a visual representation of how the function behaves for different \(x\) values.

Estimating limit as \(x\) approaches 5

Analyzing the graph, as \(x\) gets closer and closer to 5 from both sides (left and right), find where the function value \(h(x)\) is heading. That will be the estimated limit as \(x\) approaches 5.

Estimating limit as \(x\) approaches -1

Similarly, observe from the graph what value the function \(h(x)\) is heading towards as \(x\) gets closer and closer to -1 from both sides. That value will be the visually estimated limit as \(x\) approaches -1.