Question

find the limit if it exists.
lim (11x + 8)
x→ - 8

which of the following shows the correct expression after the limit properties have been applied?

a. 11· lim x+ lim 8
x→ - 8 x→ - 8

b. 11· lim x
x→ - 8

c. 11· lim x· lim 8
x→ - 8 x→ - 8

d. lim 8
x→ - 8

Explanation:

Step1: Apply limit - sum rule

The limit of a sum \( \lim_{x
ightarrow a}(f(x)+g(x))=\lim_{x
ightarrow a}f(x)+\lim_{x
ightarrow a}g(x)\). For \(y = 11x + 8\), we have \(\lim_{x
ightarrow - 8}(11x + 8)=\lim_{x
ightarrow - 8}(11x)+\lim_{x
ightarrow - 8}(8)\). Also, by the constant - multiple rule of limits \(\lim_{x
ightarrow a}(cf(x))=c\lim_{x
ightarrow a}f(x)\) where \(c = 11\) and \(f(x)=x\), so \(\lim_{x
ightarrow - 8}(11x)=11\lim_{x
ightarrow - 8}x\). And \(\lim_{x
ightarrow - 8}(8)=8\) (since the limit of a constant function \(y = k\) as \(x
ightarrow a\) is \(k\)). So the correct expression after applying limit properties is \(11\cdot\lim_{x
ightarrow - 8}x+\lim_{x
ightarrow - 8}8\).

Answer:

A. \(11\cdot\lim_{x
ightarrow - 8}x+\lim_{x
ightarrow - 8}8\)