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# Calculus Done Right

The mathematics of the continuous, through intuition not memorization.

# Three Sine Examples

We begin our study of limits by looking at the expression:

$\frac{\sin(x)}{x}$

This gives a well-defined number for every value of $$x,$$ except $$x=0.$$ So what can we say about the behavior of the expression at 0? We can't plug in 0 directly, but we can plug in very small, non-zero values of $$x.$$ And we can plug in smaller and smaller values, and see if there is a pattern in the value of the expression as $$x$$ approaches 0.

Investigate what happens to $$\frac{\sin(x)}{x}$$ as $$x$$ gets small. Try, for example, $$x=1,$$ $$x=0.1,$$ and $$x=0.01.$$ What seems to be happening to $$\frac{\sin(x)}{x}$$?

Use a calculator. There are plenty of online calculators, like this one.

Note: Make sure you are evaluating $$\sin$$ in radian mode.

We saw that as $$x$$ gets smaller and smaller, the expression $$\frac{\sin(x)}{x}$$ gets closer and closer to 1. We'll have a lot more to say about the details, but this is the basic idea behind saying "the limit as $$x$$ approaches 0 of $$\frac{\sin(x)}{x}$$ is 1." In symbols:

$\lim_{x \to 0}\frac{\sin(x)}{x} = 1$

Note that as $$x$$ gets small, both the numerator and denominator get close to 0. This limit is an example of the $$\frac{0}{0}$$ indeterminate form that we saw in the first Chapter. We’ll be encountering this form a lot.

Let's try another example. Using a calculator, let's investigate $$\lim\limits_{x \to 0} \sin\left(\frac{1}{x}\right).$$ What happens to $$\sin\left(\frac{1}{x}\right)$$ as we plug in smaller and smaller values for $$x?$$

You should have found that the expression isn't getting closer and closer to any particular number as $$x$$ approaches 0. We say that $$\lim\limits_{x \to 0} \sin\left(\frac{1}{x}\right)$$ does not exist.

Which of these arguments explains this behavior?

A) As $$x$$ gets close to 0, $$\sin(x)$$ also gets close to 0, and when we take the reciprocal of $$\sin(x)$$, we get something that gets close to infinity, so it doesn't get closer and closer to any particular number.

B) As $$x$$ gets close to 0, $$\sin(x)$$ oscillates between 0 and 1, and so does the reciprocal, so it never gets close to any particular number.

C) When you actually plug in $$x=0,$$ you get $$\sin\left(\frac{1}{0}\right),$$ which is undefined, so the limit does not exist.

D) As $$x$$ gets close to 0, $$\frac{1}{x}$$ gets bigger and bigger, so we're putting a bigger and bigger number into $$\sin,$$ and since $$\sin$$ oscillates forever between -1 and 1 as its input gets large, the final result doesn't get closer to any particular number.

Let's investigate one final example: $$\lim\limits_{x \to 0} \; x \sin\left(\frac{1}{x}\right).$$ What do you think happens to $$x \sin\left(\frac{1}{x}\right)$$ as $$x$$ gets small?

A) I think it doesn't approach any particular number, because, as we saw in the last question, the $$\sin\left(\frac{1}{x}\right)$$ term oscillates between -1 and 1 as $$x$$ gets small.

B) I think it gets closer and closer to 0, because even though $$\sin\left(\frac{1}{x}\right)$$ is doing crazy things as $$x$$ gets small, we're also multiplying by $$x,$$ and that's going to 0.

C) I think it gets closer and closer to 1, but I'm not sure why.

Let's end this quiz by getting some graphical insight. The graph of $$y = \sin\left(\frac{1}{x}\right)$$ looks like:

This is a strange graph, but we can see that as $$x$$ gets small, the corresponding $$y$$ values are alternating between $$-1$$ and $$1$$, faster and faster. So $$\displaystyle\lim_{x \to 0} \sin\left(\frac{1}{x}\right)$$ does not exist.

But the graph of $$y = x \sin\left(\frac{1}{x}\right)$$ looks like:

You can think of the $$x$$ factor as an amplitude that's changing. Instead of oscillating between 0 and 1, this expression oscillates between the lines $$y=x$$ and $$y= -x.$$ And you can see that as $$x$$ gets small, the corresponding $$y$$ values get small too. So $$\displaystyle\lim_{x \to 0} x \sin\left(\frac{1}{x}\right) = 0.$$

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