60 STEP 4. Review the Knowledge You Need to Score High
Example 2
Evaluate the limit: limx→−∞
3 x− 10
4 x^3 + 5.
Divide every term in the numerator and denominator by the highest power ofx. In thiscase, it isx^3. Thus, limx→−∞
3 x− 10
4 x^3 + 5
=xlim→−∞3
x^2−
10
x^3
4 +5
x^3=
0 − 0
4 + 0
= 0.
Verify your result with a calculator. (See Figure 5.2-4.)[– 4,4] by [–20,10]
Figure 5.2-4Example 3
Evaluate the limit: limx→∞
1 −x^2
10 x+ 7.
Divide every term in the numerator and denominator by the highest power ofx. In thiscase, it isx^2. Therefore, limx→∞
1 −x^2
10 x+ 7
=xlim→∞1
x^2− 1
10
x+
7
x^2=
xlim→∞(
1
x^2)
−xlim→∞(1)xlim→∞(
10
x)
+xlim→∞7
x^2.The limitof the numerator is−1 and the limit of the denominator is 0. Thus, limx→∞
1 −x^2
10 x+ 7=−∞.
Verify your result with a calculator. (See Figure 5.2-5.)[–10,30] by [–5,3]
Figure 5.2-5
Example 4
Evaluate the limit: limx→−∞
√^2 x+^1
x^2 + 3.
As x → −∞, x < 0 and thus, x =−√
x^2. Divide the numerator and
denominator by x (not x^2 since the denominator has a square root). Thus, you