Calculating limits

When the limit of a function is not immediately obvious, there are certain techniques we can employ to find it. It might for instance be the case that the function is a ratio of two expressions that both tend to infinity as $x$ grows large. In such a case we must consider which one of them grows faster compared to the other. One method to determine this is to compare the expressions' derivatives.

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    Johann Bernoulli (1667-1748) was a Swiss mathematician. He was the mentor to Leonhard Euler, one of the titans of maths, as well as a prominent mathematician in his own right.

    In 1694, he was offered francs by Guilllaume de l'H么pital - a member of the French aristocracy and a maths aficionado - to share all his mathematical knowledge.

    Thus, Bernoulli's rule for computing limits became known to posteriority as l'H么pital's rule.


    Some limits are evident by the construction of a function.

    However, sometimes we're faced with a situation where:


    at the given point, or as . In these cases, the limit isn't evident.

    Calculus is all about rates of change. By comparing the rates of change of the numerator and the denominator, we may still work out the limit.

    There are two hacks for calculating limits: l'H么pital's rule (sorry Bernoulli!), and Taylor's theorem.


    For example, consider the limit:

    Diagnosis: this is a 0/0 kind of limit. Medicine: l'H么pital's rule or Taylor's theorem.

    L'H么pital's rule says that:

    being a number or . Then:

    Another option is to use Taylor's theorem to expand the numerator. Recall that:

    Consequently, the limit is:

    L'H么pital's rule for limits

    So far, we've only been able to compute the most basic kind of limits. And by that, I mean limits that can be worked out by plugging in function values or after a bit of algebraic work and aid from standard limits.

    There are a couple of limits that, despite having undergone your algebraic torture, won't pop out. These are the dreaded and limits.

    In some cases, you can use a hack called L'H么pital's rule.

    Ok, it was actually discovered by the Swiss mathematician Johann Bernoulli. However, a wealthy French dude called L'H么pital payed Bernoulli to disclose all his mathematical discoveries. Hence, the rule bears L'H么pital's name.

    In fact, Bernoulli was also the first to discover the number , commonly called Euler's number, Bernoulli should get more praise!

    L'H么pital's rule says that and limits can be computed by differentiating the numerator and the denominator, assuming they're differentiable.

    You typically add the little H just to signify that you've used L'H么pital's rule. To see how this works, let us evaluate the following limit:

    Applying L'H么pital's rule once, then we get:

    Then doing so twice more gets us:

    To get to the good stuff, we had to use l'H么pital's rule a total of three times here. As you can tell from the graph, the function indeed approaches .

    Taylor expansions for limits

    So, there are basically three kinds of limits:

    • Friendly limits: Nothing strange going on. Just plug in some large values and see what happens. You'll get there by brute force or by using standard limits.

    • limits: Use l'H么pital's rule.

    • limits: Use l'H么pital's rule.

    But l'H么pital's rule isn't necessarily your best choice. You might have to compute a monstruous derivative, and you might end up with a situation again. If so, you've got to do the l'H么pital thing again. And maybe again. Continue until you squeeze out a non-zero denominator. Each time you use l'H么pital, the derivatives might get even more complicated.

    However, there's a better solution. I know it seems a bit far fetched, but we can use Taylor expansions to deal with certain limits too.

    For example, have a look at this limit:

    You might recognise this function from the section on L'H么pital's rule. There, we had to use L'H么pital's rule three times to get an answer.

    Using the Taylor expansions of the numerator and the denominator, we get:

    The last expression tends to when . And we're done. To be fair, this also involved some quite some work. Some limits are nasty, irrespective of which methods you use. But it would've been harder using l'H么pital.

    Anyway, notice how the lower order terms cancel out, and the terms of higher order than disappear. Satisfying, right?

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