# Problem evaluating a limit as x to infinity algebraically

• fedaykin
In summary, the conversation discusses finding the limit of a function and the confusion surrounding how to solve it. The person has attempted to divide and factor the equation but is unsure if their methods are valid. They are seeking help and clarification on how to prove their answer.
fedaykin
1. My problem is such:

Find the limit of $$\lim_{x \rightarrow \infty} \sqrt{9x^2+x} -3x$$

2. No relevant equations

3. I multiplied $$\frac{\sqrt{9x^2+x} -3x}{1} * \frac{\sqrt{9x^2+x} +3x}{\sqrt{9x^2+x} +3x} = \frac{x}{\sqrt{9x^2+x} +3x}$$

I am now quite confused as to where to go from here. My teacher will accept tabled values, but I wish to prove my answer. I would greatly appreciate any help.

I did attempt to divide by the highest power in the denominator, but all that got me was a mess:

$$\frac{1}{\frac{\sqrt{9x^2+x}}{x} +3}$$

Ok, I'm not certain this is valid, but...:

I'll factor out an x under the radical, then attempt to simplify

$$\frac{1}{\frac{\sqrt{9x^2*(1+\frac}{x}{9x^2}{x} +3}$$

Hmm... I'm having trouble with tex and that.

Last edited:
If you pull an x out of the square root you get

$$\frac{x}{x ( \sqrt{9 + 1/x} + 3)} = \frac{1}{ \sqrt{9+1/x} + 3}$$

which you should be able to do

Thank you so very much.

## 1. What is a limit?

A limit is a mathematical concept that describes the behavior of a function as the input approaches a certain value, usually denoted by the variable x.

## 2. How do I evaluate a limit algebraically?

To evaluate a limit algebraically, you can use techniques such as factoring, rationalization, and substitution to simplify the expression and then plug in the value that x is approaching to determine the limit.

## 3. What does it mean to evaluate a limit as x approaches infinity?

Evaluating a limit as x approaches infinity means finding the value that a function approaches as the input x gets larger and larger without bound.

## 4. Can a limit at infinity be undefined?

Yes, a limit at infinity can be undefined if the function does not approach a specific value as x gets larger. It can also be undefined if the function has a vertical asymptote at infinity.

## 5. Is there a difference between evaluating a limit at infinity and evaluating a limit at a specific value?

Yes, there is a difference. When evaluating a limit at infinity, we are interested in the overall behavior of the function as x gets larger without bound. When evaluating a limit at a specific value, we are interested in the behavior of the function as x approaches a specific value.

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