1) Problem: Evaluate the following limit (if it exists)

Solution: We try to substitute infinity. We know that when n is large, we can ignore the "+1" part (see intuitive evaluation in Theory - Limits), so we get

 

 

We see that the first part is fine, but the second needs some work. We obtained an indeterminate ratio, for which we have a box. So this is a straightforward example, we split the limit into two parts to be handled separately, then change the second part into functions and apply the l'Hospital rule.

 

 

What happens if we get seduced into putting the terms together? We get

 

 

If we had only the popular version of the l'Hospital rule, we would have to prove now that the numerator goes to infininty. This is true, since powers beat logarithms (see the scale of powers and Intuitive evaluation in Theory - Limit), and it can be proved by factoring, see the box polynomials and ratios with powers.

 

Fortunately, we also have a version of the l'Hospital rule for "something over infinity", so we can apply it right away. Check that it leads to the same answer as above.

 

 

Problem 2 : Evaluate the following limit (if it exists)

 

 

Solution: What do we get if we try to plug in infinity?  3 gives infinity, but (-2) does not have a limit. This shows that we do not get the answer by limit algebra.

So we have to try something. Since we do not have a limit for some terms, this sequence is not of any popular type of operations (like indeterminate ratio). Is it at least some kind of expression for which we have a special "box"? Fortunately yes, this example fits the box "polynomials and ratios with powers". So we know that it should be solvable by factoring dominant terms. In order to identify them we have to first simplify the given fraction:

 

 

We see that 3ⁿ is the dominant term in the numerator and 4ⁿ is the dominant term in the denominator. We factor them out:

 

 

Note how in those little fractions we pulled out n as a common exponent. This is the usual way to handle fractions of the form aⁿ/bⁿ, turning them into geometric sequences.

 

Now we are ready to evaluate the limit. Note that we have lots of geometric sequences there. All of them have the property that their bases are less than one (in absolute value) and therefore they converge to zero:

 

 

Is there any other way? Not really. Note that we cannot pass to functions and try some trick from that part (like l'Hospital), because we cannot consider in general the power (-2)^x; indeed, we only have exponentials for positive bases.

 

 

Problem 3 : Evaluate the following limit (if it exists)

 

Solution: When we try to substitute infinity, we get infinity in the denominator, which is a good start, but in the numerator we get an indeterminate expression  -  under the root, then another infinity subtracted, so we cannot really say what type we get there.

 

In fact, we get "something over infinity", which means that the more general form of l'Hospital's rule could be used (see the box "indeterminate ratio"). Note how fortunate it is that we do have this general form. If we wanted to use the usual l'Hospital (infinity over infinity), we would have to investigate the numerator and try to show that it tends to infinity, which may not be easy and perhaps it is not even true! How would we do it? Note that the fourth power under the root is dominant, and the root changes it into n², which would cancel with the second term "-n²". This shows that factoring out of the dominant term like in the box "polynomials and ratios with powers" would not help; we would need to do the cancelling properly algebraically, but the root prevents it. Fortunately, for such a problem we have a box, "difference of roots", and applying it to the numerator we would find that it goes to negative infinity, thus making the fraction into "infinity over infinity" indeed.

Anyway, thanks to our more general version of l'Hospital we can avoid this work. Still, we cannot use the l'Hospital rule yet, since it only works for ratios, but here we also have the cube on the outside (it is done last). Fortunately for us, cube is a "nice" function and we can pull it out of the limit; that is, we can ignore it, calculate the limit of the fraction inside and then cube the answer. So let's look at the fraction; we know that to use l'Hospital we have to pass to a function problem:

 

 

We obtained an expression that is not really nicer than the one we started with. This shows one typical feature of the l'Hospital rule: it is not good in getting rid of square roots. Indeed, if we tried another l'Hospital, the root would still be there, and again, and again. It is possible that after some, say, five l'Hospitals we would get an answer, but obviously this is the time to try something else. Good that we did not waste time on analyzing the numerator.

When we look at the given fraction (ignoring the cube on the outside), we notice that it features powers and roots; that is, it exactly fits the box "polynomials and ratios with powers". Could we determine the answer intuitively?

 

We actually briefly did it above, now we do it properly. We should start inside the root and observe that the fourth power prevails over the cube inside. Thus the term "-2n³" can be ignored when n is huge. What do we get then?

 

 

Unfortunately, we got two dominant terms in the numerator of the same kind and they are subtracted, which means that the intuitive way would not give an answer but leads to an indeterminate expression, in this case  - , or when simplified, ·0. Indeed, if we try to factor out the dominant term (starting with the root), we get

 

 

 

 

This type ("indeterminate product") is typically solved using the l'Hospital rule, but we tried it already and you are welcome to check that if you try it in this setting, it will be equally unpleasant.

So, what else can we do? Our trouble is in the numerator, where we essentially have the term n² - n², but we cannot subtract to see what is left because of the root. This suggest that we try to get rid of the root and there is a box exactly for that: difference of roots. So we apply the appropriate trick:

 

 

 

 

We obtained a different expression of the "ratio with powers" type, in which we were able to substract the two dominant terms in the numerator. Unless we are really unlucky, we should be now able to apply the intuitive calculations and get the answer.

 

 

 

 

So it worked, we guess that the answer is (-1)³ = -1 (we should not forget the cube!). How do we write it properly? First we pull out the cube, then do the algebra part (we can just copy the answers we already got above). Then we factor out the dominant powers (thanks to our intutitive calculation we know that the dominant power is n³ on the top and bottom, and in the denominator it comes from two sources) and finally get the answer: