Tuscany and Sophia were out hiking. As they followed the path up the side of a hill, they discovered that there had been a washout from the recent storm. The path had been obliterated for a space of about 15 feet in front of them, thereafter it continued on up the mountain from about 10 feet higher.

How could this situation be explained with one-sided limits?

### One-Sided Limits

Unlike the functions from prior lessons where the limit was the same from both directions, the functions we explore in this lesson may have a different limit for each "side." To evaluate these functions, we deal with each "side" separately, first evaluating what happens as the limit is approached from either the positive direction (the values of *x* are bigger than at the limit) or the negative direction (the values of *x* are smaller than at the limit) and then evaluating the other direction afterward as if it were effectively a separate function.

Sometimes the same value is approached from either "side" of the limit value, and some functions have different limits on the two sides of *x* = *x*_{0}.

When the value of *f*(*x*) does not get closer and closer to some *single* value *L* as \begin{align*}x \rightarrow x_0\end{align*} we say that the limit *of the function* as *x* approaches *x*_{0} *does not exist.* We may separately state the limits for each "side" of *x*_{0}, but the complete function will only have a specified limit if it is the same for both sides.

For example, the two-sided limit of the complete function \begin{align*}lim_{x \rightarrow 0} \frac{ |x|}{x}\end{align*} does not exist because the values of *f*(*x*) do not approach a single number as *x* approaches 0. We *can* state the one-sided limits for each side, since the values approach -1 from the left and 1 from the right.

### Examples

#### Example 1

Earlier, you were asked a question about how to represent a situation as a one-sided limit.

Tuscany and Sophia's path could be examined as a discontinuous function of elevation based on distance traveled along the path. For instance, if Tuscany and Sophia had traveled for 500 yards along the path before encountering the washout, then the limit of the function from the "trailhead side" would be the elevation at the edge they encountered. The function would then be undefined for the next 5 yards or so (since the path does not exist), and would pick up at 506 yards, where the elevation would be 10ft higher. If Sayber were coming **down** the path toward Tuscany and Sophia, from his point of view the "limit" of the elevation would be 10ft greater, and would be the **lowest** elevation that "his side" of the function could attain.

#### Example 2

Identify the limit of the function: \begin{align*}f(x) = \frac{|x|} {x} = \begin{cases} 1, x > 0\\ -1, x < 0\\ \end{cases} \end{align*}

which is shown in the graph below:

As *x* approaches 0 from the right, *f*(*x*) approaches 1. On the other hand, as *x* approaches 0 from the left, the function *f*(*x*) approaches -1. Since the limit is not the same from both sides, the limit of the function does not exist. However we can say that:

\begin{align*}\lim_{x \rightarrow 0^+} \frac{|x|} {x} = 1\end{align*}

\begin{align*}\lim_{x \rightarrow 0^-} \frac{|x|} {x} = -1\end{align*}

Where the superscript “+” indicates a limit from the right and the superscript “-” indicates a limit from the left.

#### Example 3

Consider the function *f* graphed below:

Find:

- \begin{align*} \lim_{x \rightarrow 2^-}f(x)\end{align*}

From graph, we can see that, \begin{align*}\lim_{x \rightarrow 2^-}f(x) = -2\end{align*}.

- \begin{align*} \lim_{x \rightarrow 2^+}f(x)\end{align*}

We can also see from the graph that \begin{align*}\lim_{x \rightarrow 2^+}f(x) = 4\end{align*}.

- \begin{align*}\lim_{x \rightarrow 2}f(x)\end{align*}

Since the limits from the right and the left are not equal (they do not approach a single value *L*), the limit does not exist. That is, \begin{align*} \lim_{x \rightarrow 2}f(x) \end{align*} does not exist.

- \begin{align*}f(2)\end{align*}

\begin{align*}f(2)=1\end{align*}

#### Example 4

Consider \begin{align*}g(x) = \frac{|x - 2|}{x - 2}|3|\end{align*} in the image below:

Find:

- \begin{align*} \lim_{x \rightarrow 2^-}f(x)\end{align*}

- \begin{align*} \lim_{x \rightarrow 2^+}f(x)\end{align*}

We can also see from the graph that \begin{align*}\lim_{x \rightarrow 2^+}f(x) = 3\end{align*}.

- \begin{align*}\lim_{x \rightarrow 2}f(x)\end{align*}

Since the limits from the right and the left are not equal (they do not approach a single value *L*), the limit does not exist. That is, \begin{align*} \lim_{x \rightarrow 2}f(x) \end{align*} does not exist.

- \begin{align*}f(2)\end{align*}

\begin{align*}f(2) = \not0\end{align*} undefined

#### Example 5

Identify the limit based on the equation:

\begin{align*}g(x)= \begin{cases} 7; x = - 5\\ 2; x \not= -5\\ \end{cases} \end{align*}

The cases specify that if \begin{align*}x = -5\end{align*} then \begin{align*}g(x) =7\end{align*} and if \begin{align*}x\end{align*} is anything else, then \begin{align*}g(x) = 2\end{align*}

\begin{align*}\therefore\end{align*} the limit as x approaches -5 from either direction is 2.

#### Example 6

Identify the limit based on the equation, use a graphing tool: \begin{align*}\lim_{x\to3^+}\frac{x^2 - 5x + 6}{x - 3}\end{align*}.

Factor the numerator: \begin{align*}(x - 2)(x - 3)\end{align*}

Now that you now have \begin{align*}(x - 3)\end{align*} in the numerator and in the denominator

Substitute 3 in for *x* in \begin{align*}x - 2\end{align*} since 3 is the number we want to evaluate

\begin{align*}3 - 2 = 1\end{align*}

\begin{align*}\therefore \lim_{x\to3^+}\frac{x^2 - 5x + 6}{x - 3} = 1\end{align*}

### Review

Identify the limit, based on each graph.

- \begin{align*}\lim_{x\to-3^-}\end{align*}
- \begin{align*}\lim_{x\to2^+}\end{align*}
- \begin{align*}\lim_{x\to-1^+}\end{align*} and \begin{align*}\lim_{x\to-1^-}\end{align*}
- \begin{align*}\lim_{x\to-1}\end{align*}
- \begin{align*}\lim_{x\to-2^-}\end{align*} and \begin{align*}\lim_{x\to5^+}\end{align*}

Identify the limit based on the equation:

- \begin{align*}\lim_{x\to2^+}\frac{-x^2 - 2x + 8}{x - 2}= \end{align*}
- \begin{align*}g(x)= \begin{cases} 4; x = - 3\\ 1; x \not= -3\\ \end{cases}\end{align*}
- \begin{align*}\lim_{x\to0^+}\frac{-x^2 + 4x}{x}= \end{align*}
- \begin{align*}g(x)= \begin{cases} -5; x \not= -1\\ -1; x = -1\\ \end{cases} \end{align*}
- \begin{align*}\lim_{x\to1^+}\frac{4x^2 - x - 3}{x - 1}= \end{align*}
- \begin{align*}f(x)= \begin{cases} 4 ; x \geq 3\\ x + 1; x < 3\\ \end{cases} \end{align*}
- \begin{align*}\lim_{x\to0^+}\frac{x^2 - 4x}{x}= \end{align*}
- \begin{align*}h(x)= \begin{cases} 4x + 4 ; x \not= 2\\ 1 ; x = 2\\ \end{cases} \end{align*}
- \begin{align*}\lim_{x\to2^-}\frac{4x^2 - 7x - 2}{x - 2}= \end{align*}
- \begin{align*}g(x)= \begin{cases} x - 5 ; x = -2\\ 4x + 1 ; x \not= -2\\ \end{cases} \end{align*}
- \begin{align*}g(x)= \begin{cases} -3x ; \not= 3\\ -9 ; x = 3\\ \end{cases} \end{align*}
- \begin{align*}\lim_{x \to -5^-}\frac{-3x^2 - 13x + 10}{x + 5}= \end{align*}
- \begin{align*}f(x)= \begin{cases} x ; x = 2\\ 3x - 3 ; x \not= 2\\ \end{cases} \end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.2.