# Find the limit as x approaches infinity

• Jan Hill
In summary, the problem is finding the limit as x approaches infinity of (the square root of x^2 + x minus the square root of x^2 -x) and the suggested method is to multiply by the conjugate, which results in an indeterminate form of [inf/inf]. The solution involves factoring out x^2 and bringing it out of the radicals, leading to a final limit of 2x.
Jan Hill

## Homework Statement

Find the limit as x approaches infinity of (the square root of x^2 + x minus the square root of x^2 -x)

## The Attempt at a Solution

I do not know how to simplify the expression. I know that plugging in x = inifinity would be wrong. How about multiplying by the conjugate i.e. (the square root of x^2 +x plus the sqaure root of x^ 2 - x

Yes, rationalize :-).

After you multiply by 1 in the form of sqrt(x^2 + x) + sqrt(x^2 - x), you will get another indeterminate form, [inf/inf]. You can deal with this one by factoring x^2 out of each term in the two radicals, and bringing it out of the radicals as x.

When I work this out, I get zero in the numerator. I don't know how to get the indeterminate form, [inf/inf] that you got. How do you get that?

I end up with 2x^2 - 2x^2 in the numerator when I myultiply by the conjugate.

"Find the limit as x approaches infinity of (the square root of x^2 + x minus the square root of x^2 -x)"

I must be missing something. Why are we talking about fractions? This is how I'm reading the problem:
$$\lim_{x \to \infty} \left( \sqrt{x^2 + x} - \sqrt{x^2 -x} \right)$$

Or is the idea to multiply this by a fraction where the numerator and denominator is the conjugate?

You're right...I don't know how I got fractions involved.

Mulitplying by the conjugate, I end up with the limit as x---> infinity of 2x

Is this right?

The fractions got involved because of the suggestion to multiply by the conjugate. Since you can't just multiply by the conjugate, but instead need to multiply by 1 in the form of the conjugate over itself, you end up with a fraction.

IOW, the original expression can be written as
$$\sqrt{x^2 + x} - \sqrt{x^2 - x} \cdot \frac{\sqrt{x^2 + x} + \sqrt{x^2 - x}}{\sqrt{x^2 + x} + \sqrt{x^2 - x}}$$

Jan Hill said:
When I work this out, I get zero in the numerator. I don't know how to get the indeterminate form, [inf/inf] that you got. How do you get that?
You have made a mistake. You should not get zero in the numerator. Check your signs.
Jan Hill said:
I end up with 2x^2 - 2x^2 in the numerator when I myultiply by the conjugate.

## What is the meaning of "limit as x approaches infinity"?

The limit as x approaches infinity refers to the value that a function approaches as the input (x) becomes infinitely large. In other words, it is the value that the function gets closer and closer to as the input increases without bound.

## Why is it important to find the limit as x approaches infinity?

Finding the limit as x approaches infinity is important because it helps us understand the behavior of a function as the input values get larger and larger. It can also help us determine if a function has an asymptote (a line that the function gets closer to but never reaches) at infinity.

## How do you find the limit as x approaches infinity?

To find the limit as x approaches infinity, you can use several methods such as substitution, factoring, or algebraic manipulation. Another method is to graph the function and observe the behavior as x gets larger. In some cases, you may need to use L'Hopital's rule or other advanced techniques.

## What are some common types of limits as x approaches infinity?

Some common types of limits as x approaches infinity include polynomials, rational functions, exponential functions, and logarithmic functions. These types of limits may have different methods for finding the limit, but the concept of approaching infinity remains the same.

## Is it possible for a function to have a limit as x approaches infinity?

Yes, it is possible for a function to have a limit as x approaches infinity. For example, the limit of a polynomial or rational function as x approaches infinity will always approach either positive or negative infinity. However, there are also cases where a function does not have a limit as x approaches infinity, such as oscillating or oscillating functions.

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