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

Precalculus

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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.

A General Note

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 \to a} f (x) = L

if and only if

\lim_{x \to a^{-}} f (x) = \lim_{x \to 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.

How To

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

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

Finding a limit using a graph

Determine the following limits and function value for the function f shown in [link] .
1. $\lim_{x \to 2^{-}} f (x)$
2. $\lim_{x \to 2^{+}} f (x)$
3. $\lim_{x \to 2} f (x)$
4. $f (2)$
Determine the following limits and function value for the function f shown in [link] .
1. $\lim_{x \to 2^{-}} f (x)$
2. $\lim_{x \to 2^{+}} f (x)$
3. $\lim_{x \to 2} f (x)$
4. $f (2)$

Looking at [link] :
1. $\lim_{x \to 2^{-}} f (x) = 8;$ when $x < 2,$ but infinitesimally close to 2, the output values get close to $y = 8.$
2. $\lim_{x \to 2^{+}} f (x) = 3;$ when $x > 2,$ but infinitesimally close to 2, the output values approach $y = 3.$
3. $\lim_{x \to 2} f (x)$ does not exist because $\lim_{x \to 2^{-}} f (x) \neq \lim_{x \to 2^{+}} f (x);$ 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 $(2, 3) .$
Looking at [link] :
1. $\lim_{x \to 2^{-}} f (x) = 8;$ when $x < 2$ but infinitesimally close to 2, the output values approach $y = 8.$
2. $\lim_{x \to 2^{+}} f (x) = 8;$ when $x > 2$ but infinitesimally close to 2, the output values approach $y = 8.$
3. $\lim_{x \to 2} f (x) = 8$ because $\lim_{x \to 2^{-}} f (x) = \lim_{x \to 2^{+}} f (x) = 8;$ 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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