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Using a graphing utility to determine a limit

With the use of a graphing utility, if possible, determine the left- and right-hand limits of the following function as x approaches 0. If the function has a limit as x approaches 0, state it. If not, discuss why there is no limit.

f ( x ) = 3 sin ( π x )

We can use a graphing utility to investigate the behavior of the graph close to x = 0. Centering around x = 0 , we choose two viewing windows such that the second one is zoomed in closer to x = 0 than the first one. The result would resemble [link] for [ 2 , 2 ] by [ 3 , 3 ] .

Graph of a sinusodial function zoomed in at [-2, 2] by [-3, 3].

The result would resemble [link] for [ −0.1 , 0.1 ] by [ −3 , 3 ] .

Graph of the same sinusodial function as in the previous image zoomed in at [-0.1, 0.1] by [-3. 3].
Even closer to zero, we are even less able to distinguish any limits.

The closer we get to 0, the greater the swings in the output values are. That is not the behavior of a function with either a left-hand limit or a right-hand limit. And if there is no left-hand limit or right-hand limit, there certainly is no limit to the function f ( x ) as x approaches 0.

We write

lim x 0 ( 3 sin ( π x ) )  does not exist .
lim x 0 + ( 3 sin ( π x ) )  does not exist .
lim x 0 ( 3 sin ( π x ) )  does not exist .
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Numerically estimate the following limit: lim x 0 ( sin ( 2 x ) ) .

does not exist

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Access these online resources for additional instruction and practice with finding limits.

Key concepts

  • A function has a limit if the output values approach some value L as the input values approach some quantity a . See [link] .
  • A shorthand notation is used to describe the limit of a function according to the form lim x a f ( x ) = L , which indicates that as x approaches a , both from the left of x = a and the right of x = a , the output value gets close to L .
  • A function has a left-hand limit if f ( x ) approaches L as x approaches a where x < a . A function has a right-hand limit if f ( x ) approaches L as x approaches a where x > a .
  • A two-sided limit exists if the left-hand limit and the right-hand limit of a function are the same. A function is said to have a limit if it has a two-sided limit.
  • A graph provides a visual method of determining the limit of a function.
  • If the function has a limit as x approaches a , the branches of the graph will approach the same y - coordinate near x = a from the left and the right. See [link] .
  • A table can be used to determine if a function has a limit. The table should show input values that approach a from both directions so that the resulting output values can be evaluated. If the output values approach some number, the function has a limit. See [link] .
  • A graphing utility can also be used to find a limit. See [link] .

Section exercises

Verbal

Explain the difference between a value at x = a and the limit as x approaches a .

The value of the function, the output, at x = a is f ( a ) . When the lim x a f ( x ) is taken, the values of x get infinitely close to a but never equal a . As the values of x approach a from the left and right, the limit is the value that the function is approaching.

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Explain why we say a function does not have a limit as x approaches a if, as x approaches a , the left-hand limit is not equal to the right-hand limit.

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Practice Key Terms 4

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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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