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Estimating a limit from a table of values near x = 0

Consider the function g(x)=sin⁡(x)xg(x) = \dfrac{\sin(x)}{x}. Note that gg is not defined at x=0x = 0, since plugging in x=0x=0 gives 00\frac{0}{0}. We want to understand what happens to g(x)g(x) as xx gets closer and closer to 00, without ever actually reaching it.

Use a calculator (in radian mode) to compute values of g(x)=sin⁡(x)xg(x) = \dfrac{\sin(x)}{x} for xx-values close to 00 from both sides, such as x=±0.5,±0.1,±0.01,±0.001x = \pm 0.5, \pm 0.1, \pm 0.01, \pm 0.001. You may organize these into a table as scratch work. The graph of gg near x=0x=0 is shown below for reference.

Estimating a limit from a table of values near x = 0 graph

Based on the trend in your values, what is lim⁡x→0sin⁡(x)x\displaystyle\lim_{x \to 0} \dfrac{\sin(x)}{x}?

Hint

Plug in smaller and smaller values of xx from both the negative and positive side, and watch what value g(x)g(x) gets closer to. Remember that g(0)g(0) itself is undefined, but that has no bearing on the limit — only the nearby values matter.

Step-by-step solution

As xx approaches 00 from either side, computing sin⁡(x)/x\sin(x)/x for x=±0.5,±0.1,±0.01,±0.001x = \pm 0.5, \pm 0.1, \pm 0.01, \pm 0.001 gives values that get closer and closer to 11 (for example, sin⁡(0.1)/0.1≈0.99833\sin(0.1)/0.1 \approx 0.99833 and sin⁡(−0.01)/(−0.01)≈0.99998\sin(-0.01)/(-0.01) \approx 0.99998). The values from the left and the values from the right both approach the same number, 11, even though g(0)g(0) itself is undefined.

Since both the left-hand and right-hand trends approach the same value, we conclude the two-sided limit exists and equals that common value. Therefore lim⁡x→0sin⁡xx=1\displaystyle\lim_{x\to 0}\frac{\sin x}{x} = 1.

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