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12.1 Finding limits: numerical and graphical approaches  (Page 4/10)

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Using the graph of the function   y = f ( x )   shown in [link] , estimate the following limits.

Graph of a piecewise function that has three segments: 1) negative infinity to 0, 2) 0 to 2, and 3) 2 to positive inifnity, which has a discontinuity at (4, 4)

a. 0; b. 2; c. does not exist; d. − 2 ;   e. 0; f. does not exist; g. 4; h. 4; i. 4

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Finding a limit using a table

Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of   x   approach   a   from both sides. Then we determine if the output values get closer and closer to some real value, the limit       L .

Let’s consider an example using the following function:

lim x →   5 ( x 3 − 125 x − 5 )

To create the table, we evaluate the function at values close to   x = 5.   We use some input values less than 5 and some values greater than 5 as in [link] . The table values show that when   x > 5   but nearing 5, the corresponding output gets close to 75. When   x > 5   but nearing 5, the corresponding output also gets close to 75.

Table shows that as x values approach 5 from the positive or negative direction, f(x) gets very close to 75. But when x is equal to 5, y is undefined.

Because

lim x → 5 − f ( x ) = 75 = lim x → 5 + f ( x ) ,

then

lim x → 5 f ( x ) = 75.

Remember that   f ( 5 )   does not exist.

Given a function   f , use a table to find the limit as   x   approaches   a   and the value of   f ( a ) , if it exists.

  1. Choose several input values that approach   a   from both the left and right. Record them in a table.
  2. Evaluate the function at each input value. Record them in the table.
  3. Determine if the table values indicate a left-hand limit and a right-hand limit.
  4. If the left-hand and right-hand limits exist and are equal, there is a two-sided limit.
  5. Replace   x   with   a   to find the value of   f ( a ) .

Finding a limit using a table

Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit.

lim x → 0 ( 5 sin ( x ) 3 x )

We can estimate the value of a limit, if it exists, by evaluating the function at values near   x = 0.   We cannot find a function value for   x = 0   directly because the result would have a denominator equal to 0, and thus would be undefined.

f ( x ) = 5 sin ( x ) 3 x

We create [link] by choosing several input values close to   x = 0 , with half of them less than   x = 0   and half of them greater than   x = 0.   Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table.

The table values indicate that when   x < 0   but approaching 0, the corresponding output nears   5 3 .

When   x > 0   but approaching 0, the corresponding output also nears   5 3 .

Table shows that as x values approach 0 from the positive or negative direction, f(x) gets very close to 5 over 3. But when x is equal to 0, y is undefined.

Because

lim x → 0 − f ( x ) = 5 3 = lim x → 0 + f ( x ) ,

then

lim x → 0 f ( x ) = 5 3 .
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Is it possible to check our answer using a graphing utility?

Yes. We previously used a table to find a limit of 75 for the function   f ( x ) = x 3 − 125 x − 5   as   x   approaches 5. To check, we graph the function on a viewing window as shown in [link] . A graphical check shows both branches of the graph of the function get close to the output 75 as   x   nears 5. Furthermore, we can use the ‘trace’ feature of a graphing calculator. By appraoching   x = 5   we may numerically observe the corresponding outputs getting close to   75.

Graph of an increasing function with a discontinuity at (5, 75)

Numerically estimate the limit of the following function by making a table:

lim x → 0 ( 20 sin ( x ) 4 x )

lim x → 0 ( 20 sin ( x ) 4 x ) = 5

Table showing that f(x) approaches 5 from either side as x approaches 0 from either side.
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Is one method for determining a limit better than the other?

No. Both methods have advantages. Graphing allows for quick inspection. Tables can be used when graphical utilities aren’t available, and they can be calculated to a higher precision than could be seen with an unaided eye inspecting a graph.

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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