Stuck Trying to find the limit of this function.

Question

We were given the function below in maths and were told to find the limit as x approaches 0

$\lim_{x\to0}(1+(2x)^2)^{1/x^2}$

the way i did it was just to make all the x's tend towards zero as follows:

$\lim_{x\to0}(1+(0)^2)^{1/0}$

Which i said that means that the limit is undefined. When we went over the answers the professor gave the answer as

$\lim_{x\to0}=e^4$

I am not quite sure how they got this, so could someone point to where i went wrong in my thinking.

Answer 1

Let $y = \frac{1}{4x^2}$, then $y\to\infty$. $$\lim_{y\to\infty}(1+\frac{1}{y})^{4y}$$ You substituted x by y in the previus step. Now $$\lim_{y\to\infty}(1+\frac{1}{y})^{4y} = \lim_{y\to \infty}((1+\frac{1}{y})^{y})^4=e^4$$ That is becaouse $e$ is defined as $\lim_{y\to \infty}(1+\frac{1}{y})^{y}$

Answer 2

$$\begin{align} & \lim_{x\to0}\left(1+(2x)^2\right)^{\frac1{x^2}}=\lim_{x\to0}\left(1+4x^2\right)^{\frac1{x^2}} \\ & \lim_{x\to\infty}\left(1+\frac4{x^2}\right)^{x^2}=\lim_{x\to\infty}\left(1+\frac4{x}\right)^{x} \\ \end{align}$$

Consider:

$$\begin{align} & e^x=\lim_{n\to\infty}\left(1+\frac{x}n\right)^n \\ \implies& \lim_{x\to0}\left(1+(2x)^2\right)^{\frac1{x^2}}=e^4 \\ \end{align}$$

Answer 3

When one theorem says a limit is undefined, like $\infty \cdot 0$, it doesn't guarantee there is not a limit, it just tells you that you need to look closer because you haven't reached a form the theorem can deal with. Compare (with $x \to 0$ from above) $$\lim_{x\to 0} \frac 1x \cdot x^2=\lim_{x\to 0}x=0\\ \lim_{x\to 0} \frac 1x \cdot x=\lim_{x\to 0}1=1 \\ \lim_{x \to 0} \frac 1{x^2} \cdot x=\lim_{x\to 0}\frac 1x=+ \infty$$ All of these are of the form $\infty \cdot 0$ The fact that we have different values on the right side is a demonstration that the basic form $\infty \cdot 0$ is not well enough defined to have an answer. Often you can cancel the product or difference of leading terms analytically, leaving a finite answer.

Answer 4

I am giving you a general form for solving above type of questions, $\lim_{y\to \infty}(1+\frac{1}{y})^{y}$=$e$ If you clearly see the function, you will realise that the function is a typical example of $1^{(\infty)}$ type of limit. Now to find the power of $e$,
Subtract $1$ from function and multiply it with the exponent of function to get the power of $e$.