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.

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 $-\infty$.
failed

Solution

failed
failed

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 \( x^2 \):

\[ \frac{9x^2 + 2x}{9x^2 - 5x + 1} = \frac{9 + \frac{2}{x}}{9 - \frac{5}{x} + \frac{1}{x^2}} \]

Step 3: Evaluate the Limit

As \( x \) approaches infinity, the terms \(\frac{2}{x}\), \(\frac{5}{x}\), and \(\frac{1}{x^2}\) approach 0. Therefore, the function simplifies to:

\[ \lim_{x \to \infty} \frac{9 + \frac{2}{x}}{9 - \frac{5}{x} + \frac{1}{x^2}} = \frac{9 + 0}{9 - 0 + 0} = \frac{9}{9} = 1 \]

Final Answer

The limit exists and is equal to 1. Therefore, the answer is:

\[ \boxed{1} \]

The correct choice is A.

Was this solution helpful?
failed
Unhelpful
failed
Helpful