Wondering if a limit exists or not

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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.
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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##
 
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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##).
 
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##\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.
 
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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##.
 
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Related to Wondering if a limit exists or not

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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