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Section 1.5

The Limit

This is section 1.5

The question: #

What happens to the $y$-values of a function as the $x$-values get closer to a particular value?

Limits: An informal definition #

We write $\displaystyle \lim_{x \to a} f(x) = L$ and say “the limit of $f(x)$, as $x$ approaches $a$, is $L$” if we can make the values of $f(x)$ arbitrarily close to $L$ (as close to $L$ as we want) by making the values of $x$ sufficiently close to $a$ but not equal to $a$.

i.e. “You want this part machined within 0.001mm error tolerance (y-values)? I can adjust my tools (x-values) to do that!”

Example #

Guess the value of $\displaystyle \lim_{x\to 1} \frac{x-1}{x^2 -1}$

note: you can’t just plug in $1$ because that function isn’t defined there.

Example #

Take a moment guess the value of $\displaystyle \lim_{x\to 0} \dfrac{\sin x}{x}$.

Note: In calculus, we’re nearly always using radians, so these $x$ values are radians. If you’re using Google Sheets, the =SIN function automatically uses radians. If you’re using a calculator (or another tool) make sure you’re in radian mode.

Click to reveal the answer. Did you guess 0? You're right!

One-sided limits #

Definition: #

We write $$ \lim_{x\to a^-} f(x) = L $$ and say the limit of $f(x)$ as $x$ approaches $a$ from the left is equal to $L$ if we can make the values of $f(x)$ arbitrarily close to $L$ (as close to $L$ as we want) by making the values of $x$ sufficiently close to $a$ but not equal to $a$.

Similarly, we define $$ \lim_{x\to a^+} f(x) = L $$ to be the right-hand limit.

Connection to total limit #

We say that: $$ \lim_{x \to a} f(x) = L \text{ if and only if } \lim_{x\to a^-} f(x) = L \text{ and } \lim_{x\to a^+} f(x) = L $$

Some visual examples #

Here’s a Geogebra applet that demonstrates visually different one-sided and two-sided limits compared to the total limit. Click ‘animate’ next to each limit and watch as the $y$-values as the $x$-values get close to the desired number.

Be sure to adjust the slider that says Example = # because that shows three different limit examples.

Note that, from the Geogebra applet:

  1. On example 1, the left and write limits are the same, so $\lim_{x\to -3} f(x) = 3$, even though the function’s value is -1 there, and
  2. On example 3, because the left and right limits aren’t the same, we say $\lim_{x\to 2} f(x)$ does not exist and often write DNE.

Infinite Limits #

Example #

Consider $\displaystyle \lim_{x\to 0} \frac1{x^2}$.

That graph obviously looks like this:

Graph of 1 over x

Looking at the left and right limits, we have:

$\lim_{x\to 0^-} \frac{1}{x^2} = \infty$ and $\lim_{x\to 0^+} \frac{1}{x^2} = \infty$

Because both sides tend to the same value, we say that limit $\displaystyle \lim_{x\to \infty} \frac{1}{x^2} = \infty$. The limit does not exist and $\infty$ is not a number, but because the left and right limits agree on the same value, we write the special value this way.

Compare to the next example

Example #

Consider the limit $\displaystyle \lim_{x\to 2} \dfrac{1}{x - 2}$.

The graph is this one:

Graph of 1 over x-2

This time, my one-sided limits are as follows:

$\displaystyle \lim_{x\to 2^-} \frac{1}{x-2} = -\infty$ while $\displaystyle \lim_{x\to 2^+} \frac{1}{x-2} = \infty$.

… in this case, the limits don’t agree, so we still say that the limit doesn’t exist but we write $\displaystyle \lim_{x\to 2} \frac{1}{x-2} = \text{DNE}$.

Visually exploring vertical asymptotes #

If you want to play with another geogebra applet, this one explores vertical asymptotes as limits. Here, you’ll click the points and move them left and right. Watch as the x-values get close to the location of the asymptote and see the y-values head up to infinity or down to negative infinity. These are infinite limits from the left and right!

Now practice! #

Head over to WebAssign and work on section 1.5. If you have questions, let me know!