4.4 Limits 99
This is an example of a removable discontinuity; we have
and the discontinuity can be removed by redefining the function to have valuey 1 = 14
whenx 1 = 12. This example is important because taking the limit in differentiation
always involves letting the denominator go to zero.
0 Exercises 7–10
The finding of limits is necessary whenever a quantity becomes indeterminate. In
addition to the case 0 2 0, the indeterminate forms most commonly met in the physical
sciences are∞ 2 ∞and∞ 1 − 1 ∞.
EXAMPLE 4.3
Both numerator and denominator tend to infinity asx 1 → 1 ∞, but the ratio remains
finite. Thus, dividing both numerator and denominator by x
2, which is allowed if
x 1 → 1 ∞, we obtain
0 Exercises 11–13
EXAMPLE 4.4
Both squared terms tend to infinity asx 1 → 10 , but the difference remains finite. By
expanding the squared terms and simplifying, we get
= 1 − 5 x
21 + 1101 → 1 10 as x 1 → 10
0 Exercises 14, 15
22222
1
3
1
44
1
x 9
x
x
x
x
x
−−
=++
− xx
x
226
1
−+
lim
x
x
x
x
x
→
−−
=
0
222
1
3
1
100
25
3
25
13
2
1
2
222x
xx
x
x
x
=
→= as→ ∞
lim
x
x
xx
→
=
∞
25
3
2
22x
x
xx
x
xx
24
2
22
2
22
−
−
=
−+
−
=+ ≠
()()
()
if