Question

Use a graphing utility to graph the function and visually estimate the limits. \(f(t)=t|t-4|\) (a) \(\lim _{t \rightarrow 4} f(t)\) (b) \(\lim _{t \rightarrow-1} f(t)\)

Short answer

From the graph, it can be visually deduced that \( \lim _{t \rightarrow 4} f(t)\) is approximately 0 and \( \lim _{t \rightarrow -1} f(t)\) is approximately 5.

Step-by-step solution

Creating the Function graph

To better visualize the behavior of the function, graph the function \(f(t)=t|t-4|\) by substituting a series of values for t into the function and then plotting these points on a graph.

Analyzing the Graph for \(t \rightarrow 4\)

Since we need to find the limit as t approaches 4, observe the y-values on the graph as t gets closer to 4 from both directions (from values slightly less than 4 and slightly greater than 4). A glance at the graph will give a visual estimate of the limit.

Analyzing the Graph for \(t \rightarrow -1\)

Similarly, for finding the limit as t approaches -1, observe how the function behaves around a t-value of -1. Look at both sides of -1 (slightly greater and slightly less) to observe the y-values.