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12.1 Finding limits: numerical and graphical approaches  (Page 3/10)

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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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