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1.2: One sided Limits and Vertical Asymptotes

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One Sided Limits

We begin our exploration of limits by taking a look at the graphs of the  following functions

• $$\displaystyle f(x)=x+1$$

• $$\displaystyle g(x)=\frac{x^2−1}{x−1}, x \ne 1$$

• $$h(x)= \left\{ \begin{array}{cc} \displaystyle \frac{x^2−1}{x−1} & \mbox{ if } x\neq 1 \\ \\ 0 & \mbox{ if } x=1 \\ \end{array} \right.$$

which are shown in Figure $$\PageIndex{1}$$. In particular, let’s focus our attention on the behavior of each graph at and around $$x=1$$.

Example $$\PageIndex{1}$$:

Consider, $$\displaystyle f(x)=x+1$$

Example $$\PageIndex{2}$$:

Consider, $$\displaystyle g(x)=\frac{x^2−1}{x−1}, x \ne 1$$,

Example $$\PageIndex{3}$$:

Consider, $$h(x)= \left\{ \begin{array}{cc} \displaystyle \frac{x^2−1}{x−1} & \mbox{ if } x\neq 1 \\ \\ 0 & \mbox{ if } x=1 \\ \end{array} \right.$$

We can now present an informal definition of one-sided limits.

Definition: One-sided Limits

We define two types of one-sided limits.

Limit from the left:

Let $$f(x)$$ be a function defined at all values in an open interval of the form $$z$$, and let $$L$$ be a real number. If the values of the function $$f(x)$$ approach the real number $$L$$ as the values of $$x$$ (where $$x<a$$) approach the number a, then we say that $$L$$ is the limit of $$f(x)$$ as $$x$$ approaches a from the left. Symbolically, we express this idea as

$\lim_{x \to a−}f(x)=L.$

Limit from the right:

Let $$f(x)$$ be a function defined at all values in an open interval of the form $$(a,c)$$, and let $$L$$ be a real number. If the values of the function $$f(x)$$ approach the real number $$L$$ as the values of $$x$$ (where $$x>a$$) approach the number $$a$$, then we say that $$L$$ is the limit of $$f(x)$$ as x approaches a from the right. Symbolically, we express this idea as

$\lim_{x \to a+}f(x)=L.$

Example $$\PageIndex{4}$$:

$$g(x)=\dfrac{|x−2|}{x−2}$$

Let us now consider the relationship between the limit of a function at a point and the limits from the right and left at that point. It seems clear that if the limit from the right and the limit from the left have a common value, then that common value is the limit of the function at that point. Similarly, if the limit from the left and the limit from the right take on different values, the limit of the function does not exist. These conclusions are summarized in Note.

Relating One-Sided and Two-Sided Limits

Let $$f(x)$$ be a function defined at all values in an open interval containing a, with the possible exception of a itself, and let L be a real number. Then,

$\lim_{x \to a}f(x)=L$

if and only if $$\displaystyle \lim_{x \to a−}f(x)=L$$ and $$\displaystyle \lim_{x \to a+} f(x)=L$$.

App: Intuitive Notion of the Limit

Limits: Intuitive Notion of the Limit - One-Sided Limits

 f(x) = f(x) = Examples1. Jump discontinuities 12. Jump discontinuities 23. Continuous4. One jump5. Hole at x = 16. Exploding 17. Exploding 28. Infinitely wiggly9. Shifted square root

Infinite Limits

Evaluating the limit of a function at a point or evaluating the limit of a function from the right and left at a point helps us to characterize the behavior of a function around a given value. As we shall see, we can also describe the behavior of functions that do not have finite limits.

We now turn our attention to $$h(x)=1/(x−2)^2$$, the third and final function introduced at the beginning of this section (see Figure(c)). From its graph we see that as the values of x approach 2, the values of $$h(x)=1/(x−2)^2$$ become larger and larger and, in fact, become infinite. Mathematically, we say that the limit of $$h(x)$$ as x approaches 2 is positive infinity. Symbolically, we express this idea as

$\lim_{x \to 2}h(x)=+∞.$

More generally, we define infinite limits as follows:

Definitions: infinite limits

We define three types of infinite limits.

Infinite limits from the left: Let $$f(x)$$ be a function defined at all values in an open interval of the form $$(b,a)$$.

i. If the values of $$f(x)$$ increase without bound as the values of x (where $$x<a$$) approach the number $$a$$, then we say that the limit as x approaches a from the left is positive infinity and we write $\lim_{x \to a−}f(x)=+∞.$

ii. If the values of $$f(x)$$ decrease without bound as the values of x (where $$x<a$$) approach the number $$a$$, then we say that the limit as x approaches a from the left is negative infinity and we write $\lim_{x \to a−}f(x)=−∞.$

Infinite limits from the right: Let $$f(x)$$ be a function defined at all values in an open interval of the form $$(a,c)$$.

i. If the values of $$f(x)$$ increase without bound as the values of x (where $$x>a$$) approach the number $$a$$, then we say that the limit as x approaches a from the left is positive infinity and we write $\lim_{x \to a+}f(x)=+∞.$

ii. If the values of $$f(x)$$ decrease without bound as the values of x (where $$x>a$$) approach the number $$a$$, then we say that the limit as x approaches a from the left is negative infinity and we write $\lim_{x \to a+}f(x)=−∞.$

Two-sided infinite limit: Let $$f(x)$$ be defined for all $$x≠a$$ in an open interval containing $$a$$

i. If the values of $$f(x)$$ increase without bound as the values of x (where $$x≠a$$) approach the number $$a$$, then we say that the limit as x approaches a is positive infinity and we write $\lim_{x \to a} f(x)=+∞.$

ii. If the values of $$f(x)$$ decrease without bound as the values of x (where $$x≠a$$) approach the number $$a$$, then we say that the limit as x approaches a is negative infinity and we write $\lim_{x \to a}f(x)=−∞.$

It is important to understand that when we write statements such as $$\displaystyle \lim_{x \to a}f(x)=+∞$$ or $$\displaystyle \lim_{x \to a}f(x)=−∞$$ we are describing the behavior of the function, as we have just defined it. We are not asserting that a limit exists. For the limit of a function f(x) to exist at a, it must approach a real number L as x approaches a. That said, if, for example, $$\displaystyle \lim_{x \to a}f(x)=+∞$$, we always write $$\displaystyle \lim_{x \to a}f(x)=+∞$$ rather than $$\displaystyle \lim_{x \to a}f(x)$$ DNE.

Example $$\PageIndex{4}$$:

Evaluate each of the following limits,

1. $$\displaystyle \lim_{x \to 0−}\frac{1}{x^2}$$

2. $$\displaystyle \lim_{x \to 0+}\frac{1}{x^2}$$

3. $$\displaystyle \lim_{x \to 0}\frac{1}{x^2}$$

a. $$\displaystyle \lim_{x \to 0−}\frac{1}{x^2}=+∞$$;

b. $$\displaystyle \lim_{x \to 0+}\frac{1}{x^2}=+∞$$;

c. $$\displaystyle \lim_{x \to 0}\frac{1}{x^2}=+∞$$

It is useful to point out that functions of the form $$f(x)=1/(x−a)^n$$, where n is a positive integer, have infinite limits as x approaches a from either the left or right (Figure $$\PageIndex{9}$$). These limits are summarized in the above definitions.

Definition: Vertical Asymptotes

Let $$f(x)$$ be a function. If any of the following conditions hold, then the line $$x=a$$ is a vertical asymptote of $$f(x)$$.

$\lim_{x \to a−}f(x)=+∞$

$\lim_{x \to a−}f(x)=−∞$

$\lim_{x \to a+}f(x)=+∞$

$\lim_{x \to a+}f(x)=−∞$

$\lim_{x \to a}f(x)=+∞$

$\lim_{x \to a}f(x)=−∞$

Example $$\PageIndex{5}$$: Behaviour of a function at different points

Use the graph of $$f(x)$$ in Figure $$\PageIndex{10}$$ to determine each of the following values:

1. $$\displaystyle \lim_{x \to −4^−}f(x)$$;$$\displaystyle \lim_{x \to −4^+}f(x)$$; $$\displaystyle\ lim_{x→−4}f(x);f(−4)$$

2. $$\displaystyle \lim_{x \to −2^−}f(x$$);$$\displaystyle \lim_{x \to −2^+}f(x)$$; $$\displaystyle \lim_{x→−2}f(x);f(−2)$$

3. $$\displaystyle \lim_{x \to 1^−}f(x)$$; $$\displaystyle \lim_{x \to 1+}f(x)$$; $$\displaystyle \lim_{x \to 1}f(x);f(1)$$

4. $$\displaystyle \lim_{x \to 3^−}f(x)$$; $$\displaystyle \lim_{x \to 3+}f(x)$$; $$\displaystyle \lim_{x \to 3}f(x);f(3)$$

Figure $$\PageIndex{10}$$: The graph shows $$f(x)$$.

Solution

Using the definitions above and the graph for reference, we arrive at the following values:

1. $$\displaystyle \lim_{x \to −4^−}f(x)=0$$; $$\displaystyle \lim_{x \to −4^+}f(x)=0$$; $$\displaystyle \lim_{x \to −4}f(x)=0;f(−4)=0$$

2. $$\displaystyle \lim_{x \to −2^−}f(x)=3$$; $$\displaystyle \lim_{x \to −2^+}f(x)=3$$; $$\displaystyle \lim_{x \to −2}f(x)=3;f(−2)$$ is undefined

3. $$\displaystyle \lim_{x \to 1^−}f(x)=6$$;$$\displaystyle \lim_{x \to 1^+}f(x)=3$$; $$\displaystyle \lim_{x \to 1}f(x)$$ DNE; $$f(1)=6$$

4. $$\displaystyle \lim_{x \to 3^−}f(x)=−∞$$;$$\displaystyle \lim_{x \to 3^+}f(x)=−∞$$;$$\displaystyle \lim_{x \to 3}f(x)=−∞$$;$$f(3)$$ is undefined

.

Example $$\PageIndex{6}$$: Einstein’s Equation

Given Einstein’s equation for the mass of a moving object

$m=\dfrac{m_0}{\sqrt{1−\frac{v^2}{c^2}}}, \nonumber$

what is the value of this bound?

Figure $$\PageIndex{11}$$. (Credit:NASA)

Solution

Our starting point is Einstein’s equation for the mass of a moving object,

$m=\dfrac{m_0}{\sqrt{1−\frac{v^2}{c^2}}}, \nonumber$

where $$m_0$$ is the object’s mass at rest, $$v$$ is its speed, and $$c$$ is the speed of light. To see how the mass changes at high speeds, we can graph the ratio of masses $$m/m_0$$ as a function of the ratio of speeds, $$v/c$$ (Figure $$\PageIndex{11}$$).

Figure $$\PageIndex{11}$$: This graph shows the ratio of masses as a function of the ratio of speeds in Einstein’s equation for the mass of a moving object

Glossary

infinite limit
A function has an infinite limit at a point a if it either increases or decreases without bound as it approaches a
intuitive definition of the limit
If all values of the function $$f(x)$$ approach the real number L as the values of $$x(≠a)$$ approach a, $$f(x)$$ approaches L
one-sided limit
A one-sided limit of a function is a limit taken from either the left or the right
vertical asymptote
A function has a vertical asymptote at $$x=a$$ if the limit as x approaches a from the right or left is infinite