Calculating the Limit of a Function without use of L'Hospital's Rule or Power Series

Question

I came across this question the other day and have been trying to solve it by using some simple algebraic manipulation without really delving into L'Hospital's Rule or the Power Series as I have just started learning limit calculations. We needed to find : $$\lim_{x \to 0} \frac {x\cos x - \sin x}{x^2\sin x}$$ I approached this problem in two different ways and know what the flaw is, however I have been unable to justify why this is so.

Let $$f(x) = \frac {x\cos x - \sin x}{x^2\sin x}$$ Therefore, dividing by $x$, $$f(x) = \frac {\cos x - \frac{\sin x}{x}}{x\sin x}$$ Using standard limit properties, $$\lim_{x \to 0}f(x) = \frac{\lim_{x \to 0}\cos x - \lim_{x \to 0}\frac{\sin x}{x}}{\lim_{x \to 0}x\sin x}$$ Since $$\lim_{x \to 0} \frac {\sin x}{x}=1$$ $$\lim_{x \to 0}f(x)= \frac{\lim_{x \to 0}\cos x-1}{\lim_{x \to 0}x\sin x}$$

Rewriting the above as $$\lim_{x \to 0}\frac{(\cos x -1)x}{x^2\sin x}$$ and using the fact that $\lim_{x \to 0} \frac {\sin x}{x}=1$ and $\lim_{x \to 0}\frac{\cos x -1}{x^2}= -\frac{1}{2}$, we get $$\lim_{x \to 0}f(x)=-\frac{1}{2}$$

I know that the answer is wrong although I am not able to understand why. I believe it is because I cannot combine the numerator and denominator into a single limit function. Using a similar trick, I also obtained the limit to be $-\frac{3}{8}$.

Questions:

1) Could someone please explain why combining the numerator and denominator into a single limit is wrong? (The reason I even went ahead with such a manipulation was, we are allowed to separate the numerator and denominator while expanding the limit of a rational function so I felt that the reverse should also work).

2) As you can notice, I have not used L'Hospital's Rules or Power Series expansion of $\sin x $and $\cos x$. When I used L'Hospital's Rule, I noticed that I needed to go upto the third or fourth derivative to get rid of the $\frac{0}{0}$ indeterminate form. So would there be a better way of approaching such limits?

Thank You.

Answer 1

First, note that both of your answers are incorrect: the actual answer is $\frac{-1}{3}$.

For your questions:

1. The problem is actually earlier than that: it is not necessarily the case that $\lim \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)}$ unless both limits exist, and $\lim g(x) \neq 0$. In this case, that latter condition isn't satisfied, so this method does not work. In particular, when you write:

$$\lim_{x\to 0}f(x) = \dfrac{\lim_{x\to 0}\cos(x) - \lim_{x\to 0}\frac{\sin(x)}{x}}{\lim_{x\to 0} x\sin(x)},$$

what you've actually written is $\lim\limits_{x\to 0}f(x) = \frac{\mbox{[something]}}{0}$, which is obviously nonsense, and so everything after that is necessarily flawed.

2. L'Hôpital's rule (he's not a hospital!) is how I'd approach this. There might be a trick to do it quickly, but it's not obvious if there is.

Answer 2

Your approach does not use the laws of limits in proper manner. Each of those laws which deal with limit of sum/difference/product/quotient of two functions has certain restrictions which must be checked before applying them.

In the current question the issue is with application of the quotient rule. You should note that the rule applies only when the limit of denominator is non-zero. But here limit of denominator $x\sin x$ is $0$ and thus your first step is wrong.

Further note that the question does require the use of L'Hospital's Rule or Taylor series. Using the limit $\lim\limits _{x\to 0}\dfrac{\sin x} {x} =1$ you can replace the denominator $x^2\sin x$ (as given in original problem) by $x^3$ and the numerator should be split as $$x\cos x-x+(x-\sin x) $$ so that the fraction under limit can be replaced by $$\frac{x-\sin x} {x^3}-\frac{1-\cos x} {x^2}$$ The second fraction tends to $1/2$ (via algebraic manipulation) and the first one tends to $1/6$ (via L'Hospital's Rule or Taylor series) and thus the desired limit is $1/6-1/2=-1/3$.

Answer 3

1) If I understand your question, you are conjecturing that:

$$ \lim_{x \to a} \frac{f(x)}{g(x)}= \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)} $$

This is wrong. You may try to prove it, but as a simple counterexample should suffice:

Let $f(x)=x$ and $g(x) =x$. Then: $$ \lim_{x \to a} \frac{f(x)}{g(x)}= \lim_{x \to a} \frac{x}{x} = 1 $$ And the result holds even for $a=0$. but: $$ \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}= \frac{a}{a} $$ And the last one is undetermined for $a =0$, since $a/a$ is not within a limit expression anymore. This would actually hold if ${\lim_{x \to a} g(x)}\neq 0$, but this exception was precisely the case you had in hand.

2) You have a rather complicate expression there. I don't see any elegant/short solution to it, but using power series seems pretty much fair and straight forward.

There could be some property related to limits and some known transform such as Initial value theorem from Laplace's transform, but that wouldn't enter into my book as a "better" solution.