How Do You Compute a Limit From a Graph?

In the next several examples, a graph is used to evaluate one-sided limits from the right and left, the corresponding two-sided limit, and the value of the function. The key is to find the y values as the x values approach some value from the left and the right…if those match, the two sided limit is the y value that it matches at.

Does anyone see any mistakes in the examples above?

How Do You Calculate a Limit Algebraically?

You can recognize the limits by what happens when you substitute the value x approaches into the expression. If it gives 0/0, there is algebra that you can do to find the exact value of the limit.

In the first two examples, the expression may be factored and simplified…then you can substitute the value for x.

$\underset{x\to 3}{\mathop{lim }}\frac{{{x}^{2}}-5x+6}{x-3}$

$\underset{x\to -1}{\mathop{lim }}\frac{{{x}^{2}}-x-2}{x+1}$

In the next two examples, the fractions in the numerator must be combined before the fraction may be simplified.

$\underset{x\to 0}{\mathop{lim }}\frac{\frac{1}{x-6}+\frac{1}{6}}{x}$

$\underset{x\to 0}{\mathop{lim }}\frac{\frac{1}{4}-\frac{1}{x+4}}{x}$

The next two examples are designed to throw you off. When you substitute the value into the expression, you do not get 0/0. This means you need to use a table or graph to get the limit.

$\underset{x\to -5}{\mathop{lim }}\frac{1}{{{\left( x+5 \right)}^{2}}}$

$\underset{x\to 1}{\mathop{lim }}\frac{x}{{{\left( x-1 \right)}^{2}}}$

The next two examples show how to rationalize the numerator to do a limit.

$\underset{x\to 1}{\mathop{lim }}\frac{\sqrt{x}-1}{x-1}$

This example may be done two different ways as the next two boards demonstrate.

$\underset{x\to 4}{\mathop{lim }}\frac{\sqrt{x}-2}{x-4}$

How Do You Compute the Limit of a Difference Quotient?

The last part of Section 10.3 asks you to compute several different difference quotients. Some of the problems ask you to compute

$\underset{h \to 0}{\mathop{lim }},\frac{f(a+h)-f(a)}{h}$

where f(x) and a are given to you in the problem. Here are a few examples from the board.

$f(x)=4x+3$ and $a=1$

The board above contains a mistake…do you see where this group made a mistake?

$f(x)={{x}^{2}}-4$ and $a=1$

$f(x)={{x}^{2}}-1$ and $a=2$

In the examples below, you are asked to compute a difference quotient containing x instead of a.

Compute $\underset{h \to 0}{\mathop{lim }},\frac{f(x+h)-f(x)}{h}$ where $f(x)={{x}^{2}}+2x$

Compute $\underset{h \to 0}{\mathop{lim }},\frac{f(x+h)-f(x)}{h}$ where $f(x)={{x}^{2}}-x$

By the way, the correct solution to the first problem is below.

In this original calculation, f(a+h) and f(a) where switched.

How Do You Find the Average Rate of Change from a Table?

One of the problems on the homework gave you three points on a line graph, (1905, 1024), (1955, 240), (2005, 1141). In these ordered pairs, the x value is the year and the y value is the number of immigrants (in thousands) to a large country.

1. Find the average rate of change in immigration from 1905 to 1955 in immigrants per year.
2. Find the average rate of change in immigration from 1955 to 2005 in immigrants per year.
3. Find the average rate of change in immigration from 1905 to 2005 in immigrants per year.

This problem illustrates the two ways that you can work in the “thousands” in the data to give immigrants per year instead of thousands of immigrants per year.

How Do You Find the Average Rate of Change From a Function?

Problem 1 Find the average rate of change of $\displaystyle f(x)=ln (x)$ over [2, 4] to four decimal places.

Problem 2 Find the average rate of change of $\displaystyle f(x)= {e}^{x}$ over [1, 3] to four decimal places.

When you calculate the rate to four decimal places, you should write the numbers in the quotient to FIVE decimal places to make sure there are no rounding errors.