Page 72 - Calculus Demystified

CHAPTER 2
Foundations of Calculus
calculate lim x→2 ), we can then divide the denominator of the expression 59
defining g into the numerator. Thus
g(x) = x + 2 for x = 2.
Now
lim g(x) = lim x + 2 = 4.
x→2 x→2

Fig. 2.2

The graph of the function g is shown in Fig. 2.2. We encourage the reader to
use a pocket calculator to calculate values of g for x near 2 but unequal to 2 to
check the validity of our answer. For example,

2
g(x) =[x − 4]/[x − 2]
x
1.8 3.8
1.9 3.9
1.99 3.99
1.999 3.999
2.001 4.001
2.01 4.01
2.1 4.1
2.2 4.2

We see that, when x is close to 2 (but unequal to 2), then g(x) is close (indeed,
as close as we please) to 4.

3
2
x − 3x + x − 3
You Try It: Calculate the limit lim x→3 .
x − 3
Math Note: It must be stressed that, when we calculate lim x→c f(x),we do not
evaluate f at c. In the last example it would have been impossible to do so. We want
to determine what we anticipate f will do as x approaches c, not what value (if any)
f actually takes at c. The next example illustrates this point rather dramatically.