Question

find the limit if it exists.
lim(x→ - 1)(10x + 9)
which of the following shows the correct expression after the limit properties have been applied?
a. 10·lim(x→ - 1)x
b. 10·lim(x→ - 1)x·lim(x→ - 1)9
c. lim(x→ - 1)9
d. 10·lim(x→ - 1)x+lim(x→ - 1)9

Explanation:

Step1: Apply limit - sum property

The limit of a sum $\lim_{x
ightarrow a}(f(x)+g(x))=\lim_{x
ightarrow a}f(x)+\lim_{x
ightarrow a}g(x)$. Here $f(x) = 10x$ and $g(x)=9$. So $\lim_{x
ightarrow - 1}(10x + 9)=\lim_{x
ightarrow - 1}(10x)+\lim_{x
ightarrow - 1}(9)$.

Step2: Apply constant - multiple limit property

The limit of a constant - multiple $\lim_{x
ightarrow a}(cf(x))=c\lim_{x
ightarrow a}f(x)$ for a constant $c$. For $c = 10$ and $f(x)=x$, $\lim_{x
ightarrow - 1}(10x)=10\lim_{x
ightarrow - 1}x$. So $\lim_{x
ightarrow - 1}(10x + 9)=10\lim_{x
ightarrow - 1}x+\lim_{x
ightarrow - 1}9$.

Answer:

D. $10\cdot\lim_{x
ightarrow - 1}x+\lim_{x
ightarrow - 1}9$