Figuring out Limits

I have a set of limits questions, which I’m really confused with!!

1. Why does (X^2)*e^X go to infinity as x tends to infinity ? Do we have to use l’hopital’s ?

2. Why does cos(x)/(x-pi/2)^2 go to infinity at pi/2? I thought it would be indeterminate and then it would go to 0 using l’hopitals

3. Why does (x^2 + 2x)/x^5 go to infinity as x tends to 0? I thought using l’hopitals we’d end up with 6x leading to 0 as the limit.

Any help on these 3 would be much appreciated!!

1. No need for L'Hopitals. I'd say this should be very obvious. Both terms in the product, $x^2$ and $e^x$, individually tends to $\infty$ so of course the product will as well.

2. You can apply L'Hopital's rule here once and see that this limit is equivalent to $\dfrac{-\sin x}{2(x-\pi/2)}$ as $x\to \pi/2$. The numerator approaches a finite value but the denominator tending to zero makes the whole limit shoot off to infinity.

3. One application of L'Hopital's lands you on $\displaystyle \lim_{x \to 0} \dfrac{2x + 2}{5x^4}$. The numerator tends to the finite value of 2, but the denominator going to zero makes the whole limit shoot off to infinity.

This is really useful thank you

I think I’m mainly getting confused with the idea of the denominator being 0- I thought this meant it’s indeterminate so I just did the l’hopitals again and again

How do you tell whether it’s diverging or whether to do l’hopitals again?

This is really useful thank you

I think I’m mainly getting confused with the idea of the denominator being 0- I thought this meant it’s indeterminate so I just did the l’hopitals again and again

How do you tell whether it’s diverging or whether to do l’hopitals again?

And one other tiny question, for the first question, if x tends to minus infinity, what happens ?

You should check what an indeterminate form looks like again. After one application of L'Hopital you lose the indeterminate form so you cannot apply L'Hopital again.

That last question doesn't make much sense. You can tell whether a quotient is diverging or not by seeing whether either the numerator grows to infinity faster than the denominator, and/or the denominator itself tends to zero.

So does this mean if it’s going to infinity, it’s essentially diverging? So would question 2 and 3 be diverging at those values?

I understand the point about l’hopitals now- so once one value is finite, it’s no longer indeterminate - have I got this right?

I’m also wondering what happens if in the first question x tends to minus infinity- using l’hopitals ?

I have a set of limits questions, which I’m really confused with!!

1. Why does (X^2)*e^X go to infinity as x tends to infinity ? Do we have to use l’hopital’s ?

2. Why does cos(x)/(x-pi/2)^2 go to infinity at pi/2? I thought it would be indeterminate and then it would go to 0 using l’hopitals

3. Why does (x^2 + 2x)/x^5 go to infinity as x tends to 0? I thought using l’hopitals we’d end up with 6x leading to 0 as the limit.

Any help on these 3 would be much appreciated!!