# Wondering if a limit exists or not

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• mcastillo356
In summary: In this case, the limit is said to be ##\pm \infty##, but again, it is considered not to exist.]In summary, the conversation discusses the limit ##\lim_{x \to{-}\infty}{(\sqrt{x^2+x}-x)}## and whether or not it exists. While some may argue that the limit is ##\infty##, it is generally agreed that the limit does not exist since infinity is not considered a number in the real number system. Additionally, the limit can also fail to exist if different values are obtained on either side of the limit point.
mcastillo356
Gold Member
TL;DR Summary
My textbook says the limit does not exist. I don't agree, or there is something I miss.
I have opposite conclusions about ##\lim_{x \to{-}\infty}{(\sqrt{x^2+x}-x)}##

Quote from my textbook:
"The limit ##\lim_{x \to{-}\infty}{(\sqrt{x^2+x}-x)}## is not nearly so subtle. Since ##-x>0## as ##x\rightarrow{-\infty}##, we have ##\sqrt{x^2+x}-x>\sqrt{x^2+x}##, which grows arbitrarily large as ##x\rightarrow{-\infty}##. The limit does not exist."

But online limits calculators say the limit is ##\infty##, and my personal opinion is:
##\lim_{x \to{-}\infty}{(\sqrt{x^2+x}-x)}=\infty-(-\infty)=\infty##

For ##x < 0##,\begin{align*}
\lim_{x \rightarrow -\infty} \sqrt{x^2 + x} - x &= \lim_{x \rightarrow -\infty} |x| \left(\sqrt{1+\dfrac{1}{x}} + 1 \right) \\
&= \lim_{x \rightarrow -\infty} |x| \left( 2 + \dfrac{1}{2x} + O\left( \dfrac{1}{x^2} \right) \right) \\
&= \lim_{x \rightarrow -\infty} 2|x| - \dfrac{1}{2}
\end{align*}which does not exist (i.e. you can make the thing on the right arbitrarily large for sufficiently negative ##x##).

mcastillo356
##\pm \infty ## are not numbers. A sequence is divergent if it increases or decreases forever to ##\pm \infty ##. It is not convergent to infinity. In this sense, the limit does not exist.

Last edited:
mcastillo356
mcastillo356 said:
[My textbook says the limit does not exist. I don't agree, or there is something I miss.
It's sort of a semantics thing. Although we can write ##\lim_{x \to \infty}x^2 = \infty##, if the "limit" is ##\infty##, we consider the limit to not exist, since ##\infty## is not a number in the real number system.

Another way that the limit can fail to exist is if you get different values on either side of the limit point, as in ##\lim_{x \to 0} \frac 1 x##.

mcastillo356

## 1. What is a limit in mathematics?

A limit in mathematics is a fundamental concept used to describe the behavior of a function as its input approaches a certain value. It is used to determine the value that a function approaches as its input gets closer and closer to a specific value, without actually reaching it.

## 2. How do you know if a limit exists or not?

A limit exists if the function approaches a single value as the input approaches a specific value. This means that the left-hand and right-hand limits must be equal at that point. If the left-hand and right-hand limits are not equal, then the limit does not exist.

## 3. What are the different types of limits?

There are three types of limits: finite limits, infinite limits, and limits at infinity. A finite limit is when the function approaches a single value as the input approaches a specific value. An infinite limit is when the function approaches either positive or negative infinity as the input approaches a specific value. A limit at infinity is when the function approaches a single value as the input approaches positive or negative infinity.

## 4. How do you determine if a limit is finite or infinite?

To determine if a limit is finite or infinite, you need to evaluate the left-hand and right-hand limits at the specific value. If both limits are equal and finite, then the limit is finite. If one or both limits are infinite, then the limit is infinite.

## 5. Why is it important to determine if a limit exists or not?

Determining if a limit exists or not is important because it helps us understand the behavior of a function at a specific point. It also allows us to solve problems involving rates of change, continuity, and other real-world applications. Additionally, knowing if a limit exists or not can help us determine the convergence or divergence of a series, which is important in calculus and other areas of mathematics.

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