The limit laws

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We give basic laws for working with limits.

Video Lecture

Compute the following limits (Hint: If you need to, you can try graphing each of the functions to see what is happening at $x=\pi$ .)

$\lim _{x\to \pi } x^3=\answer {\pi ^3},\quad \lim _{x\to \pi } \sqrt {2}=\answer {\sqrt {2}},\quad \lim _{x\to \pi } e^x= \answer {e^{\pi }},$

Whether you did it analytically or by using a graph, the key to computing the above limits is that the functions are all continuous at $\pi$ . But what about the following limits?

$\begin{align*} &\lim_{x\to \pi} (x^3-e^{x}),\\ &\lim_{x\to \pi} \frac{\sqrt{2}}{e^{x}},\\ &\lim_{x\to \pi} (\sqrt{2}\cdot x^3\cdot e^{x}), \text{ or}\\ &\lim_{x\to \pi} e^{x^3}? \end{align*}$

We can’t use continuity here, because we don’t know if the functions $x^3-e^{x}$ , $\frac {\sqrt {2}}{e^{x}}$ , $\sqrt {2}\cdot x^3\cdot e^{x}$ , and/or $e^{x^3}$ are continuous at $x=\pi$ , and we have no other tools available, since the graphs and tables are not reliable for functions this complicated. Obviously, we need more tools to help us with computation of limits.

In this section, we present a handful of rules, called the Limit Laws, that allow us to find limits of various combinations of functions.

Limit Laws Suppose that $k\in \mathbb {R}$ , $\lim _{x\to a}f(x)=L$ , and $\lim _{x\to a}g(x)=M$ .

Constant Multiple Law: $\lim _{x\to a} kf(x) = k\lim _{x\to a}f(x)=kL$ .
Sum/Difference Law: $\lim _{x\to a} (f(x) \pm g(x)) = \lim _{x\to a}f(x) \pm \lim _{x\to a}g(x)=L \pm M$ .
Product Law: $\lim _{x\to a} (f(x)g(x)) = \lim _{x\to a}f(x)\cdot \lim _{x\to a}g(x)=LM$ .
Quotient Law: If $M\ne 0$ , then $\lim _{x\to a} \frac {f(x)}{g(x)} = \frac {\lim _{x\to a}f(x)}{\lim _{x\to a}g(x)}=\frac {L}{M}$ .

In plain language, “the limit of a sum equals the sum of the limits,” “the limit of a product equals the product of the limits,” etc.

Let’s examine how the Limit Laws can be used in computation of limits.

Compute the following limits using Limit Laws:

(a): $\lim _{x\to \pi } (x^3-e^{x})$
(b): $\lim _{x\to \pi } \frac {\sqrt {2}}{e^{x}}$
(c): $\lim _{x\to \pi } (\sqrt {2}\cdot x^3\cdot e^{x})$
(d): $\lim _{x\to \pi } e^{x^3}$

For $\lim _{x\to \pi } (x^3-e^{x})$ , write:

$\begin{align*} \lim_{x\to \pi} (x^3-e^{x})&=\lim_{x\to \pi} x^3-\lim_{x\to \pi}e^{x} && \text{by Difference Law}\\ &=\pi^3-e^{\pi} \end{align*}$

For $\lim _{x\to \pi } \frac {\sqrt {2}}{e^{x}}$ , write:

$\begin{align*} \lim_{x\to \pi} \frac{\sqrt{2}}{e^{x}}&=\frac{\lim_{x\to \pi}\sqrt{2}}{\lim_{x\to \pi}e^{x}} && \text{by Quotient Law}\\ &=\frac{\sqrt{2}}{e^{\pi}} \end{align*}$

For $\lim _{x\to \pi } (\sqrt {2}\cdot x^3\cdot e^{x})$ , using the Product Law, we can write:

$\lim _{x\to \pi } (\sqrt {2}\cdot x^3\cdot e^{x})=\lim _{x\to \pi } \sqrt {2}\cdot \lim _{x\to \pi }x^3\cdot \lim _{x\to \pi }e^{x}$

$\begin{align*} &=\sqrt{2}\cdot \pi^3\cdot e^{\pi}\\ &=\sqrt{2}\pi^3e^{\pi} \end{align*}$

For $\lim _{x\to \pi } e^{x^3}$ , write:

$\lim _{x\to \pi } e^{x^3}= \cdots$ The function $e^{x^3}$ is a combination of the functions $e^{x}$ and $x^3$ , but it is neither a sum/difference, nor a product, nor a quotient of these two functions, so we cannot apply any of the Limit Laws. This function is a composition of the two functions, $e^{x}$ and $x^3$ .

Compute the following limit using limit laws:

$\lim _{x\to 1}(5x^2+3x-2)$

Well, get out your pencil and write with me:

$\lim _{x\to 1} (5x^2+3x-2) = \lim _{x\to 1} 5x^2 + \lim _{x\to 1} \answer [given]{3x} - \lim _{x\to 1}2$

by the Sum/Difference Law. So now

$= 5\lim _{x\to 1} x^2 + 3\lim _{x\to 1} x - \lim _{x\to 1}\answer [given]{2}$

by the Product Law. Finally by continuity of $x^b$ and $k$ ,

$= 5(1)^2 + 3(1) - 2 =\answer [given]{6}.$ We can check our answer by looking at the graph of $y=f(x)$ :

$\graph {5x^2+3x-2}$

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