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Estimating a limit numerically near x = 0

Consider the function f(x)=1−cos⁡xx2f(x) = \dfrac{1 - \cos x}{x^2}. Note that f(x)f(x) 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 f(x)f(x) as xx gets very close to 00, without actually plugging in x=0x = 0.

Use a calculator or computer to evaluate f(x)=1−cos⁡xx2f(x) = \dfrac{1-\cos x}{x^2} at values of xx close to 00 from both sides, such as x=±0.1,±0.01,±0.001,±0.0001x = \pm 0.1, \pm 0.01, \pm 0.001, \pm 0.0001. Use these values to determine the trend in f(x)f(x) as xx approaches 00 from the left and from the right.

Estimating a limit numerically near x = 0 graph

Based on your table of values (and the graph), what value does f(x)f(x) appear to approach as x→0x \to 0? State lim⁡x→0f(x)\lim_{x \to 0} f(x).

Hint

Build a small table with xx-values approaching 00 from both the negative and positive side, computing f(x)f(x) for each. Look at where the outputs seem to be settling, not at any single value.

Step-by-step solution

Plugging in values like x=±0.1,±0.01,±0.001x = \pm 0.1, \pm 0.01, \pm 0.001 into f(x)=1−cos⁡xx2f(x) = \dfrac{1-\cos x}{x^2} gives outputs that get closer and closer to 0.50.5 as xx shrinks toward 00 from either side. For example, f(0.1)≈0.4996f(0.1) \approx 0.4996 and f(−0.1)≈0.4996f(-0.1) \approx 0.4996, and this value keeps approaching 0.50.5 as xx gets smaller in magnitude.

Since the values of f(x)f(x) approach the same number 0.50.5 whether xx approaches 00 from the left or from the right, and f(x)f(x) does not need to be defined at x=0x=0 itself for the limit to exist, we conclude the two-sided limit exists and equals 0.50.5. Thus lim⁡x→01−cos⁡xx2=0.5\lim_{x \to 0} \dfrac{1-\cos x}{x^2} = 0.5.

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