Question

find the limit.

$$\lim_{x \to 2} \frac{x + 7}{x + 9}$$

select the correct choice below and, if necessary, fill in the answer box to complete your choice.

a. $$\lim_{x \to 2} \frac{x + 7}{x + 9}=$$ (simplify your answer.)
b. the limit does not exist.

Explanation:

Step1: Substitute \(x = 2\) into the function

We use the direct - substitution property for limits. If \(f(x)=\frac{x + 7}{x+9}\), then \(\lim_{x
ightarrow a}f(x)=f(a)\) when \(f(a)\) is well - defined.
Substitute \(x = 2\) into \(\frac{x + 7}{x+9}\), we get \(\frac{2+7}{2 + 9}\).

Step2: Simplify the expression

Calculate the numerator \(2 + 7=9\) and the denominator \(2+9 = 11\). So, \(\frac{2+7}{2 + 9}=\frac{9}{11}\).

Answer:

A. \(\lim_{x
ightarrow2}\frac{x + 7}{x+9}=\frac{9}{11}\)