# Why Doesn't the Limit of $$xe^{-\frac{1}{x}}$$ Exist as $$x \to 0$$?

• MHB
• Vali
In summary, the limit of $xe^{-\frac{1}{x}}$ as x approaches 0 does not exist because the limits from below and above are not equal. The limit from below is 0, while the limit from above is $-\infty$.
Vali
Why the following limit doesn't exists ?
$$\lim_{x\rightarrow 0}xe^{-\frac{1}{x}}$$
I think it's because of $\frac{1}{x}$ which doesn't exists, right ?

Vali said:
Why the following limit doesn't exists ?
$$\lim_{x\rightarrow 0}xe^{-\frac{1}{x}}$$
I think it's because of $\frac{1}{x}$ which doesn't exists, right ?

the limits from both sides of zero are not equal

$\lim_{x \to 0^+} x \cdot e^{-1/x} = 0$

$\lim_{x \to 0^-} x \cdot e^{-1/x} = -\infty$

Vali said:
Why the following limit doesn't exists ?
$$\lim_{x\rightarrow 0}xe^{-\frac{1}{x}}$$
I think it's because of $\frac{1}{x}$ which doesn't exists, right ?
No. As skeeter said, it is because the limits "from below" and "from above" are not the same. If the problem were $$\lim_{x\rightarrow 0^+}xe^{-\frac{1}{x}}$$ it would still be true that "the limit, as x goes to 0, of $\frac{1}{x}$ does not exist" but as $-\frac{1}{x}$ goes to negative infinit, $e^{-\frac{1}{x}}$ goes to 0. $$\lim_{x\rightarrow 0^+}xe^{-\frac{1}{x}}= 0$$.

## 1. Why can't we just take the limit of the function?

Limits are used to determine the behavior of a function as the input approaches a certain value. However, if the function approaches different values from the left and right sides of the input, the limit does not exist.

## 2. How do we know when a limit does not exist?

A limit does not exist if the function approaches different values from the left and right sides of the input, or if the function approaches infinity or negative infinity.

## 3. Can a limit not exist at a specific point?

Yes, a limit can fail to exist at a specific point if the function has a discontinuity or a vertical asymptote at that point.

## 4. What does it mean when a limit does not exist?

When a limit does not exist, it means that the function does not have a defined value at that point. This could be due to various reasons, such as the function being undefined or approaching different values from different directions.

## 5. Is it possible for a limit to not exist even if the function is continuous?

Yes, it is possible for a limit to not exist even if the function is continuous. This can happen if the function approaches different values from the left and right sides of the input, or if the function has a vertical asymptote at that point.

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