Questions: Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
lim as x approaches infinity of (9x^2 + 2x) / (9x^2 - 5x + 1)
Select the correct choice below and, if necessary, fill in the answer box within your choice.
A. lim as x approaches infinity of (9x^2 + 2x) / (9x^2 - 5x + 1) = □
(Simplify your answer. Type an integer or a fraction.)
B. The limit does not exist and is neither infinity nor -infinity.
Transcript text: Use the properties of limits to help decide whether each limit exists. If a limit exists, find its value.
\[
\lim _{x \rightarrow \infty} \frac{9 x^{2}+2 x}{9 x^{2}-5 x+1}
\]
Select the correct choice below and, if necessary, fill in the answer box within your choice.
A. $\lim _{x \rightarrow \infty} \frac{9 x^{2}+2 x}{9 x^{2}-5 x+1}=$ $\square$
(Simplify your answer. Type an integer or a fraction.)
B. The limit does not exist and is neither $\infty$ nor $-\infty$.
Solution
Solution Steps
Step 1: Identify the Highest Degree Terms
To find the limit of the function as \( x \) approaches infinity, we first identify the highest degree terms in both the numerator and the denominator.
The given function is:
\[
\frac{9x^2 + 2x}{9x^2 - 5x + 1}
\]
The highest degree term in both the numerator and the denominator is \( 9x^2 \).
Step 2: Simplify the Function
We simplify the function by dividing both the numerator and the denominator by \( x^2 \):