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Understanding two-sided limits

In the previous example, the left-hand limit and right-hand limit as x approaches a are equal. If the left- and right-hand limits are equal, we say that the function f ( x ) has a two-sided limit    as x approaches a . More commonly, we simply refer to a two-sided limit as a limit. If the left-hand limit does not equal the right-hand limit, or if one of them does not exist, we say the limit does not exist.

The two-sided limit of function as x Approaches a

The limit of a function f ( x ) , as x approaches a , is equal to L , that is,

lim x a f ( x ) = L

if and only if

lim x a f ( x ) = lim x a + f ( x ) .

In other words, the left-hand limit of a function f ( x ) as x approaches a is equal to the right-hand limit of the same function as x approaches a . If such a limit exists, we refer to the limit as a two-sided limit. Otherwise we say the limit does not exist.

Finding a limit using a graph

To visually determine if a limit exists as x approaches a , we observe the graph of the function when x is very near to x = a . In [link] we observe the behavior of the graph on both sides of a .

Graph of a function that explains the behavior of a limit at (a, L) where the function is increasing when x is less than a and decreasing when x is greater than a.

To determine if a left-hand limit exists, we observe the branch of the graph to the left of x = a , but near x = a . This is where x < a . We see that the outputs are getting close to some real number L so there is a left-hand limit.

To determine if a right-hand limit exists, observe the branch of the graph to the right of x = a , but near x = a . This is where x > a . We see that the outputs are getting close to some real number L , so there is a right-hand limit.

If the left-hand limit and the right-hand limit are the same, as they are in [link] , then we know that the function has a two-sided limit. Normally, when we refer to a “limit,” we mean a two-sided limit, unless we call it a one-sided limit.

Finally, we can look for an output value for the function f ( x ) when the input value x is equal to a . The coordinate pair of the point would be ( a , f ( a ) ) . If such a point exists, then f ( a ) has a value. If the point does not exist, as in [link] , then we say that f ( a ) does not exist.

Given a function f ( x ) , use a graph to find the limits and a function value as x approaches a .

  1. Examine the graph to determine whether a left-hand limit exists.
  2. Examine the graph to determine whether a right-hand limit exists.
  3. If the two one-sided limits exist and are equal, then there is a two-sided limit—what we normally call a “limit.”
  4. If there is a point at x = a , then f ( a ) is the corresponding function value.

Finding a limit using a graph

  1. Determine the following limits and function value for the function f shown in [link] .
    1. lim x 2 f ( x )
    2. lim x 2 + f ( x )
    3. lim x 2 f ( x )
    4. f ( 2 )
    Graph of a piecewise function that has a positive parabola centered at the origin and goes from negative infinity to (2, 8), an open point, and a decreasing line from (2, 3), a closed point, to positive infinity on the x-axis.
  2. Determine the following limits and function value for the function f shown in [link] .
    1. lim x 2 f ( x )
    2. lim x 2 + f ( x )
    3. lim x 2 f ( x )
    4. f ( 2 )
    Graph of a piecewise function that has a positive parabola from negative infinity to 2 on the x-axis, a decreasing line from 2 to positive infinity on the x-axis, and a point at (2, 4).
  1. Looking at [link] :
    1. lim x 2 f ( x ) = 8 ; when x < 2 , but infinitesimally close to 2, the output values get close to y = 8.
    2. lim x 2 + f ( x ) = 3 ; when x > 2 , but infinitesimally close to 2, the output values approach y = 3.
    3. lim x 2 f ( x ) does not exist because lim x 2 f ( x ) lim x 2 + f ( x ) ; the left and right-hand limits are not equal.
    4. f ( 2 ) = 3 because the graph of the function f passes through the point ( 2 , f ( 2 ) ) or ( 2 , 3 ) .
  2. Looking at [link] :
    1. lim x 2 f ( x ) = 8 ; when x < 2 but infinitesimally close to 2, the output values approach y = 8.
    2. lim x 2 + f ( x ) = 8 ; when x > 2 but infinitesimally close to 2, the output values approach y = 8.
    3. lim x 2 f ( x ) = 8 because lim x 2 f ( x ) = lim x 2 + f ( x ) = 8 ; the left and right-hand limits are equal.
    4. f ( 2 ) = 4 because the graph of the function f passes through the point ( 2 , f ( 2 ) ) or ( 2 , 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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