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Unit 1

1.4 Estimating Limit Values from Tables

2 min readโ€ขjune 7, 2020

Anusha Tekumulla


Using tables to estimate limit values helps us better visualize what a limit actually is. In order to understand this concept, letโ€™s look at an example.ย ๐Ÿ’ญ

Example Problem โ“

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The easiest way to find a limit is to plug in the given x value into the function. However, youโ€™ll find that the given function isnโ€™t defined at x = 3 because the denominator evaluates to 0. Because we canโ€™t find the function value at x = 3, we find the limit by approaching x = 3.ย 

1) We have to make a table and the first step is picking the x values to use. Pick a value that's a little bit less than x = 3 (that is, a value that's "to the left" ๐Ÿ‘ˆ of 3), so maybe start with something like x = 2.9.

x

2.9

3

f(x)

0.16949

undefined

2) Next, add a couple more x-values to your table to simulate the feeling of getting infinitely close to x = 3, from the left.

x

2.9

2.99

2.999

3

f(x)

0.16949

0.16694

0.16669

undefined

3) Approach x = 3 from the right just like we did from the left.ย 

x

2.9

2.99

2.999

3

3.001

3.01

3.1

f(x)

0.16949

0.16694

0.16669

undefined

0.16664

0.16639

0.16393

Now that we have our table, we can estimate the limit of f(x) = x - 3 โ„ x^2 - 9 at x = 3 is 0.1667 or 1 โ„ 6.ย 

It is important to pick x values that get infinitely close to the target value. In the example above, we picked numbers that were closer and closer to x = 3. We wouldnโ€™t be able to estimate the limit value if we used numbers with equal increments like 2.25, 2.5, and 2.75.ย 

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