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Estimating a limit numerically from a table of values

Consider the function f(x)=x2+x−6x−2f(x) = \dfrac{x^2 + x - 6}{x - 2} Note that ff is undefined at x=2x = 2, since substituting x=2x = 2 gives 00\frac{0}{0}. We want to understand what happens to f(x)f(x) as xx gets close to 22, without plugging in x=2x = 2 itself.

Build a table of values of f(x)f(x) for xx approaching 22 from both sides — for example, use x=1.9,1.99,1.999x = 1.9, 1.99, 1.999 and x=2.1,2.01,2.001x = 2.1, 2.01, 2.001. Compute f(x)f(x) at each of these values (remember, ff is not defined at x=2x=2 itself, so skip that point). Based on the trend you observe as xx gets closer to 22 from the left and from the right, what value does f(x)f(x) appear to be approaching? In other words, what is lim⁡x→2f(x)\displaystyle\lim_{x \to 2} f(x)?

Estimating a limit numerically from a table of values graph
Hint

Plug values very close to 2 (but not equal to 2) into f(x)f(x) using a calculator, and watch what number the outputs get closer and closer to as xx approaches 2 from each side. Try factoring the numerator first to simplify the computation, if you like.

Step-by-step solution

Choosing xx-values approaching 22 from the left, such as x=1.9,1.99,1.999x = 1.9, 1.99, 1.999, and computing f(x)f(x) at each, you should find the outputs are approximately 4.9,4.99,4.9994.9, 4.99, 4.999 — getting closer and closer to 55.

Similarly, choosing xx-values approaching 22 from the right, such as x=2.1,2.01,2.001x = 2.1, 2.01, 2.001, the outputs come out to approximately 5.1,5.01,5.0015.1, 5.01, 5.001 — also closing in on 55.

Since f(x)f(x) approaches the same value, 55, from both the left and the right as x→2x \to 2, the table suggests that the two-sided limit exists and equals 55, even though f(2)f(2) itself is undefined.

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