1.5: The Limit of a Function

Goals:

The Table Method

We start with an example.

For the function $f(x) = x^2 - x + 2$, investigate what happens near $x = 2$ but not at $x = 2$.
Suppose $f(x)$ is defined when $x$ is near the number $a$. Then the notation \[\lim_{x\rightarrow a} f(x) = L\] means we can make the values of $f(x)$ arbitrarly close to $L$ by taking $x$ sufficiently close to $a$ but never, ever $a$ itself.

I think about it like this: the heights $f(x)$ can be made as close to $L$ as possible when you move $x$ close to $a$ but never $a$ itself.

This results in three possible graphs:

Our first tool for finding limits was described in the first example; it's called the table method. Create two tables that samples $x$ values to the left and right of $a$ but never $a$. Do not plug in $a$!!!!

Then see what the values of $f(x)$ are approaching. If both tables agree then the limit exists.

Use the table method to guess the following limits:
  1. $\displaystyle\lim_{x\rightarrow 1} \dfrac{x-1}{x^2 - 1}$
  2. $\displaystyle\lim_{x\rightarrow 1} g(x)$ where \[g(x) = \begin{cases} \dfrac{x-1}{x^2 - 1} & x \neq 1 \\ 2 & x = 1 \end{cases}\]
  3. $\displaystyle \lim_{x\rightarrow 0} \dfrac{\sin x}{x}$
  4. $\displaystyle\lim_{x\rightarrow 0} \left(x^3 + \dfrac{\cos 5x}{10000}\right)$

These examples show that the table method has a big pitfall: potential inaccuracy if you don't pick $x$ close enough to $a$.

One-Sided Limits

The table method requires two tables: one with $x$ values less than $a$, and another with $x$ values greater than $a$.

Here's limit notation for each table:

One-Sided Limits

The function $f$ has the right-hand limit $L$ as $x\rightarrow a$ from the right, written \[\lim_{x\rightarrow a^+} f(x) = L\] if $f(x)$ can be made close to $L$ as we please by taking $x$ sufficiently close to and to the right of $a$.

Similarly, the function $f$ has the left-hand limit $L$ as $x\rightarrow a$ from the left, written \[\lim_{x\rightarrow a^-} f(x) = L\] if $f(x)$ can be made close to $L$ as we please by taking $x$ sufficiently close to and to the left of $a$.

If left-and right-hand limits agree, the limit exists. This is equivalent to both tables agreeing. This idea is summarized as:

\[\lim_{x\rightarrow a^+}f(x) = L = \lim_{x\rightarrow a^-}f(x) \qquad \text{if and only if} \qquad \lim_{x\rightarrow a}f(x) = L\]

This theorem is very important. If left- and right-hand limits disagree, the limit does not exist.

For the Heaviside function \[H(t) = \begin{cases} 0 & t < 0 \\ 1 & t \geq 0\end{cases}\] determine if $\displaystyle\lim_{t\rightarrow 0}H(t)$ exists.

The Graph Method

Limits can also be seen on the graph: look at what the heights of the function are approaching on both sides of $x = a$.

A function $g(x)$ has the following graph: Determine if the following exist. If they do, find the value.

Infinite Limits

The symbol $\infty$ is not a number. Certainly not a real number.

This is why in interval notation you always see $(-\infty, 3)$ or $(0, \infty)$, but never $(0, \infty]$.

Mathematicians use the symbol $\infty$ to represent the idea that a quantity grows forever.

Let $f$ be a function defined on both sides of $a$, except possibly at $a$. We write \[\lim_{x\rightarrow a} f(x) = \infty\] if $f(x)$ grows without bound by taking $x$ sufficiently close but not equal to $a$.

If $\lim_{x\rightarrow a}f(x) = -\infty$, then $f(x)$ is decreasing without bound.

These conditions are equivalent to the existence of a vertical asymptote. See pictures in class.

The definition also applies to left- and right-hand limits, i.e. $\lim_{x\rightarrow a^-} f(x) = \infty$ means the heights of the graph are growing without bound as $x$ approaches $a$ from the left.

Find \[\lim_{x\rightarrow 0} \dfrac{1}{x^2}\]
Find $\displaystyle\lim_{x\rightarrow 3^+} \dfrac{2x}{x-3}$ and $\displaystyle\lim_{x\rightarrow 3^-} \dfrac{2x}{x-3}$.
Show $x = \dfrac{\pi}{2}$ is a vertical asymptote of $f(x) = \tan(x)$.

In summary, we know two ways to find limits: