Evaluating Limits from a Graph
Finding limits from the graph of a piecewise function
Evaluating a Limit Numerically
Author: ShaNisaa RaSun
The following problem is solved in this video. It is recommended that you try to solve the problem before watching the video. You can click "Answer" to reveal the answer to the problem.
Problem: Estimate \(\displaystyle {\lim_{x\to 1} \dfrac{\dfrac{1}{x}-1}{x-1}}\) numerically, if it exists.
To find the limit numerically, you need to construct a table similar to the following.
From the table, it looks like the \(y\)-values are approaching \(-1\) as \(x\) approaches 1 from the left and from the right, so that's the solution to the limit.
\(\displaystyle {\lim_{x\to 1^-} \dfrac{\dfrac{1}{x}-1}{x-1}} = -1\)
| \(x \rightarrow 1^-\) | \({f(x) =\dfrac{\dfrac{1}{x}-1}{x-1}}\) | \(x \rightarrow 1^+\) | \({f(x) =\dfrac{\dfrac{1}{x}-1}{x-1}}\) |
|---|---|---|---|
| 0.9 | -1.1111 | 1.1 | -0.9091 |
| 0.99 | -1.0101 | 1.01 | -0.9901 |
| 0.999 | -1.0010 | 1.001 | -0.9990 |
| 0.9999 | -1.0001 | 1.0001 | -0.9999 |
From the table, it looks like the \(y\)-values are approaching \(-1\) as \(x\) approaches 1 from the left and from the right, so that's the solution to the limit.
\(\displaystyle {\lim_{x\to 1^-} \dfrac{\dfrac{1}{x}-1}{x-1}} = -1\)
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