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.