Question

PLEASE PLEASE PLEASE HELP!
find the limit of the function algebraically. limx->0 x^2+3/x^4

Answer 1

3/0 --> infinity

Answer 2

show me the steps please

Answer 3

To find the limit algebraically, you must first be familiar with this:\[\Large\rm \color{orangered}{\lim_{x\to0}\frac{1}{x}=\infty}\]As x get's closer to 0, 1/x grows and grows.

\[\Large\rm \lim_{x\to0}\frac{x^2+3}{x^4}\]So what we'll do is, we'll divide the top and bottom by the largest factor of x.
Err I think dividing by the largest factor of x that shows up in the denominator ends up being sufficient. But that doesn't matter in this problem since our largest power is in the denominator anyway.\[\Large\rm \lim_{x\to0}\frac{x^2+3}{x^4}\left(\frac{\frac{1}{x^4}}{\frac{1}{x^4}}\right)=\lim_{x\to0}\frac{\frac{1}{x^2}+\frac{3}{x^4}}{1}\]
Ignore the 1 in the bottom, then rewrite this as the sum of two limits,\[\Large\rm =\lim_{x\to0}\frac{1}{x^2}+\lim_{x\to0}\frac{3}{x^4}\]Then we'll pull the junk out of the way to relate this to the thing I stated at the beginning.\[\Large\rm =\left(\color{orangered}{\lim_{x\to0}\frac{1}{x}}\right)^2+3\left(\color{orangered}{\lim_{x\to0}\frac{1}{x}}\right)^4\]So from here you can see that it's exploding as x approaches 0.\[\Large\rm =\left(\color{orangered}{\infty}\right)^2+3\left(\color{orangered}{\infty}\right)^4=\infty\]Those last couple of steps were maybe a little overkill.
You can probably see what's going on directly after you do the division.

Answer 4

These are not as much fun to do algebraically.
It's more fun to just look at it and think about what's going on.
The numerator is a 2nd degree polynomial, like a porabola.
The denominator is a 4th degree polynomial, grows and shrinks much faster than 2nd degree.

So the denominator is getting to zero much much faster.
And then you can relate it to the form that cow was talking about.