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Unit 1

1.4 Estimating Limit Values from Tables

2 min readโ€ขjune 7, 2020

anusha

Anusha Tekumulla


AP Calculus AB/BCย โ™พ๏ธ

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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“

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreenshot%20(430).png?alt=media&token=f6d4fe53-d86c-4ec9-8bb3-cb5d8993b1d6
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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