Limits

Calculus wouldn't exist without the concept of limits. It turns out that it's pretty tricky to define a limit properly, but you can get an intuitive understanding of limits even without going into the glory details

The basic idea

We start with some function \(f\) and a point on the \(x\)-axis which we call \(a\). Here is what we'd like to understand:

What does \(f(x)\) look like when \(x\) is really close to \(a\), but not equal to \(a\) ?

Suppose that we have a a function \(f\) with domain \(\mathbb{R}\backslash\{2\}\), and set \(f(x) = x - 1\) on this domain. Formally we wright:

\[ f(x) = x - 1 \quad when \quad x \ne 2 \]

The graph of the function is like:

You can get as close as you want to \(1\), without actually getting to \(1\) by letting \(x\) be close enough to \(2\). Without getting bogged down, we just write:

\[ \lim_{x \to 2}f(x) = 1 \]

which read as:

the limit, as \(x\) goes to \(2\), of \(f(x)\) is equal to \(1\). As \(x\) journeys along the number line from the left or the right toward the number \(2\), the value of \(f(x)\) gets very very close to \(1\)(and stays close!).

And what if the function has special value at limit point? For example, we have:

\[ g(x) = \begin{cases} x - 1 & if \quad x \ne 2, \\ 3 & if \quad x = 2. \end{cases} \]

What is \(\lim_{x \to 2}g(x)\)? The tricky here is that the value of \(g(x)\) is irrenlevent with \(\lim_{x \to 2}g(x)\), it's only the values of \(g(x)\) where \(x\) is close to \(2\), not actually at 2, which matter.

So, \(\lim_{x \to 2}g(x) = 1\) as before, even though \(g(x) = 3\).

Left-hand and right-hand limits

If we have a function graph like this:

Of course \(h(x) = 2\) is irrelevant as far as the limiting behavior is concerned.

Imagine that you're the hiker in the picture, climbing up and down the hill. The value of \(h(x)\) tells you how high up you are when your horizontal position is at \(x\). So if you walk right ward from the left of the picture, you get the left-hand limit of \(h(x)\) at \(x = 3\) is equal to \(1\). On the other hand, if you are walking leftward from the right-hand side of the picutre, your height becomes close to \(-2\) as your horizontal position gets close to \(x =3\). This means that the right-hand limit is equal to \(-2\).

We can summarize our findings from above by writing: $$ \lim_{x \to 3^-} h(x) = 1 $$

and

\[ \lim_{x \to 3^+} h(x) = -2 \]

The minus sign after \(3\) means the limit is a left-hand limit, and the plus sign means the limit is a right-hand limit.

The regular two-sided limit at \(x = a\) exists exactly when both left-hand and right-hand limits at \(x = a\) exist and are equal to each other. If the left-hand limit and right-hand limit are not equal, the two-sided limit does not exist, written as:

\[ \lim_{x \to 3} h(x) = DNE \]

Limits at \(\infty\) and \(-\infty\)

In above parts we've concentrated on the behavior of a function near a point \(x = a\), however sometimes it's important to understand how a function behaves when \(x\) gets really huge. We defines the large and small number as this:

A number is

• large, if its absolute value is really big number;
• small, if it is really close to \(0\)(but not actually equal to \(0\)).

We can say the limit at infinity as:

• "f has a right-hand horizontal asymptote at y = L", means:

\[ \lim_{x \to \infty} f(x) = L \]

• "f has a left-hand horizontal asymptote at y = M", means:

\[ \lim_{x \to -\infty} f(x) = M \]

Two common misconceptions about asymptotes

A function doesn't have to have the same horizontal asymptote on the left as on the right.

For example, the graph of \(y = tan^{-1}(x)\) looks like:

it has the limit of:

• \(\lim_{x \to \infty} tan^{-1}(x) = \frac{\pi} {2}\)
• \(\lim_{x \to -\infty} tan^{-1}(x) = -\frac{\pi} {2}\)

A function can cross its asymptote.

The graph of \(y = \frac{\sin(x)}{x}\) is like:

and it crosses its aymptote many times.

The sandwich principle

The sandwich principle, also known as squeeze principle, says that:

If a function \(f\) is sandwiched between two functions \(g\) and \(h\) that converge to the same limit \(L\) as \(x \to a\), then \(f\) also converges to \(L\) as \(x \to a\).

There is a similar version of the sandwich principle for one-sided limits, except this time the inequality \(g(x) < f(x) < h(x)\) only has to hold for \(x\) on the side of \(a\) that you care about.

For example, the function \(f(x) = x sin(\frac{1}{x})\) is sandwiched by \(h(x) = x\) and \(g(x) = -x\):

we can say that:

\[ \lim_{x \to +0} x \sin(\frac{1}{x}) = 0. \]

In summary if:

\[ g(x) \le f(x) \le h(x) \]

for all \(x\) near \(a\), and

\[ \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \]

then:

\[ \lim_{x \to a} f(x) = L. \]