2.2 The limit of a function  (Page 2/12)

We have calculated the values of $f\left(x\right)=(\text{sin}x)\text{/}x$ for the values of x listed in [link] .

Note : The values in this table were obtained using a calculator and using all the places given in the calculator output.

As we read down each $\frac{\left(\text{sin}x\right)}{x}$ column, we see that the values in each column appear to be approaching one. Thus, it is fairly reasonable to conclude that $\underset{x\to 0}{\text{lim}}\frac{\text{sin}x}{x}=1.$ A calculator-or computer-generated graph of $f\left(x\right)=\frac{\left(\text{sin}x\right)}{x}$ would be similar to that shown in [link] , and it confirms our estimate.

As before, we use a table—in this case, [link] —to list the values of the function for the given values of x .

After inspecting this table, we see that the functional values less than 4 appear to be decreasing toward 0.25 whereas the functional values greater than 4 appear to be increasing toward 0.25. We conclude that $\underset{x\to 4}{\text{lim}}\frac{\sqrt{x}-2}{x-4}=0.25.$ We confirm this estimate using the graph of $f\left(x\right)=\frac{\sqrt{x}-2}{x-4}$ shown in [link] .

Despite the fact that $g\left(\mathrm{-1}\right)=4,$ as the x -values approach −1 from either side, the $g\left(x\right)$ values approach 3. Therefore, $\underset{x\to \mathrm{-1}}{\text{lim}}g\left(x\right)=3.$ Note that we can determine this limit without even knowing the algebraic expression of the function.