Worksheet for Section 1.8

Section 1.8 introduces limits (the fundamental concept on which the rest of calculus is based). Recall the following function from the last section:

$$h(x)=\frac{x}{\sqrt{x+1}-1}$$

The notation used to describe the behavior of $h(x)$ near $x=0$ looks like this:

$$\lim_{x\to 0} \frac{x}{\sqrt{x+1}-1}=2$$

(Read this: ``the limit as $x$ approaches 0 of $h(x)$ is 2.'')

The formal definition of a limit, in all of its hideous detail, looks like this:

$$\lim_{x\to c} f(x)=L$$
means
``for every $\epsilon>0$, there is a $\delta>0$, such that
if $0<\vert x-c\vert<\delta$, then $f(x)-L\vert<\epsilon$''

I will show you how to use this definition to prove a fairly simple limit:

$$\lim_{x\to 2} (3x+1) = 7$$

This will be enough to show you why the definition is rarely used in practice.

Rather than the imprecise graphical and numerical approaches or the messy and tedious formal definition, it will be easier to use a few basic rules (which can be proven from the definition -- I will do this for a select few of the rules) to find limits. Here is a list of the most important rules:

Properties:

• $$\lim_{x\to c} k = k$$
• $$\lim_{x\to c} x = c$$
• $$\lim_{x\to c} [k\cdot f(x)] = k\cdot \lim_{x\to c} f(x)$$
• $$\lim_{x\to c} [f(x)\pm g(x)] = \lim_{x\to c} f(x)\pm\lim_{x\to c} g(x)$$
• $$\lim_{x\to c} [f(x)\cdot g(x)] = \lim_{x\to c} f(x)\cdot\lim_{x\to c} g(x)$$
• $$\lim_{x\to c} \left[\frac{f(x)}{g(x)}\right] = \frac{\lim_{x\to c} f(x)}{\lim_{x\to c} g(x)}$$, provided $\lim_{x\to c} g(x)\neq 0$

These rules are enough to handle a lot of algebraic functions, such as polynomials and rational functions (as long as the denominator doesn't approach zero). Here are a couple of examples -- in each case, find the limit:

$$\lim_{x\to -1} 3x-1\qquad \lim_{x\to 1} 2x^2-3x+1\qquad \lim_{x\to 2} \frac{x^2+3x-10}{x-1}$$

There is a special notation used for one-sided limits, where the limit process is carried out just from one side of the number $c$ (there is even an $\epsilon$-$\delta$ definition of one-sided limits, but we won't go there). Here is the notation:

• Limit from the Right: $$\lim_{x\to c^+} f(x)=L$$
• Limit from the Left: $$\lim_{x\to c^-} f(x)=L$$

For example, find $$\lim_{x\to 3^-}\sqrt{9-x^2}$$. The one-sided limits give an important characterization of when ordinary limits exist:

$$\lim_{x\to c} f(x)=L$$   if and only if both$$\quad \lim_{x\to c^+} f(x)=L$$   and$$\lim_{x\to c^-} f(x)=L$$

Here are some examples of limits that fail to exist -- there are several ways this can happen, and these examples show the more common ways:

$$\lim_{x\to 0}\frac{\vert x\vert}{x}\qquad \lim_{x\to 0}\frac1{x^2}\qquad \lim_{x\to 0}\sin\frac1x$$

Again, in each case graph the function and see if you can tell what is happening in the limit from the graph. A table of values may also help. (These also correspond to examples from the last section.)

For another example, graph the following two functions:

$$f(x)=\frac{1}{x^2+1}\qquad\qquad g(x)=\frac{2x^2}{x^2+1}$$

Both of these functions have horizontal asymptotes. To describe this sort of behavior, use a variation on the limit notation:

   left:$$\ \lim_{x\to-\infty} f(x)=L$$   right:$$\ \lim_{x\to\infty} f(x)=L$$

There is also an $\epsilon$-$\delta$ definition of this sort of limit, but we won't be using it. A horizontal asymptote of a function, then, is a horizontal line $y=L$, where either $$\ \lim_{x\to-\infty} f(x)=L$$ or $$\ \lim_{x\to\infty} f(x)=L$$.

Now you can write a formal definition of continuity in terms of limits -- a function $f(x)$ is continuous at $c$ if all of the following three conditions are met:

• $f(c)$ is defined
• $$\lim_{x\to c} f(x)$$ exists
• $$\lim_{x\to c} f(x) = f(c)$$

If $f(x)$ is continuous at $c$ for every $c$ in an open interval $(a,b)$, then $f(x)$ is continuous on $(a,b)$; if $f(x)$ is continuous at $c$ for every real number $c$, then $f(x)$ is everywhere continuous (sometimes simply called continuous). In the following examples, graph the functions and use the graphs to find any discontinuities (i.e. places where the functions are not continuous).

$$f_1(x)=e^x\qquad g_1(x)=3x^2-2x+5\qquad h_1(x)=\frac1{x^2}$$

$$f_2(x)=\frac{x^2+3x-10}{x-2}\qquad g_2(x)=\frac{\vert x\vert}{x}\qquad h_2(x)=\begin{cases}1-x\qquad&x\leq 0\\ x^2-1&x>0 \end{cases}$$

Note that most of the important functions from precalculus are continuous: all polynomials, $\sin x$, $\cos x$, and $e^x$ are all continuous everywhere; $\sqrt{x}$ and $\ln x$ both have restricted domains, but are continuous on those domains; $\tan x$ has discontinuities associated with vertical asymptotes, but is continuous at all other points; and rational functions are continuous except where the denominator is zero.

Also, one-sided limits can be used to fill in endpoints in the definition of continuity on an interval, as needed for the Intermediate Value Theorem in the last section: $f(x)$ is continuous on the closed interval $[a,b]$ if it is continuous on the open interval $(a,b)$ and the following one-sided limits hold:

$$\lim_{x\to a^+} f(x)=f(a)$$   and$$\lim_{x\to b^-} f(x)=f(b)$$

For example, on what closed interval is the function $f(x)=\sqrt{9-x^2}$ continuous?

Another interesting fact about functions makes it possible to extend the above rules a bit farther.

• Composition: If $$\lim_{x\to c} g(x) = M$$ and $$\lim_{x\to M} f(x) = f(M)$$ (watch for this new role $M$ is playing here), then $$\lim_{x\to c} f(g(x)) = f(M)$$.

Here are a few examples using this property -- in each case, find the limit:

$$\lim_{x\to 0}\sqrt{x+1}\qquad \lim_{x\to 0}\cos(x^2)\qquad \lim_{x\to 1} e^{x^2-1}$$