## Section 3.1 The Limit

¶The value a function \(f\) approaches as its input \(x\) approaches some value is said to be the limit of \(f\text{.}\) Limits are essential to the study of calculus and, as we will see, are used in defining continuity, derivatives, and integrals.

Consider the function

Notice that \(x=1\) does not belong to the domain of \(f(x)\text{.}\) Regardless, we would like to know how \(f(x)\) behaves close to the point \(x=1\text{.}\) We start with a table of values:

It appears that for values of \(x\) close to \(1\) we have that \(f(x)\) is close to \(2\text{.}\) In fact, we can make the values of \(f(x)\) as close to \(2\) as we like by taking \(x\) sufficiently close to \(1\text{.}\) We express this by saying *the limit of the function \(f(x)\) as \(x\) approaches \(1\) is equal to \(2\)* and use the notation:

###### Definition 3.1. Limit (Useable Definition).

In general, we write

if we can make the values of \(f(x)\) arbitrarily close to \(L\) by taking \(x\) to be sufficiently close to \(a\) (on either side of \(a\)) but not equal to \(a\text{.}\)

We read the expression \(\lim\limits_{x\to a}f(x)=L\) as “the limit of \(f(x)\) as \(x\) approaches \(a\) is equal to \(L\)”. When evaluating a limit, you are essentially answering the following question: What number does the function *approach* while \(x\) gets closer and closer to \(a\) (but *not equal* to \(a\))? The phrase *but not equal to \(a\)* in the definition of a limit means that when finding the limit of \(f(x)\) as \(x\) approaches \(a\) we never actually consider \(x=a\text{.}\) In fact, as we just saw in the example above, \(a\) may not even belong to the domain of \(f\text{.}\) All that matters for limits is what happens to \(f\) close to \(a\text{,}\) not necessarily what happens to \(f\) at \(a\text{.}\)

Interactive Demonstration. Investigate the limit \(\ds\lim_{x\to 1} \dfrac{x^2-1}{x-1} \) using the sliders below:

### Subsection 3.1.1 One-sided limits

Consider the following piecewise defined function:

Observe from the graph that as \(x\) gets closer and closer to \(1\) from the*left*, then \(f(x)\) approaches \(+1\text{.}\) Similarly, as \(x\) gets closer and closer \(1\) from the

*right*, then \(f(x)\) approaches \(+2\text{.}\) We use the following notation to indicate this:

The symbol \(x\to 1^-\) means that we only consider values of \(x\) sufficiently close to \(1\) which are less than \(1\text{.}\) Similarly, the symbol \(x\to 1^+\) means that we only consider values of \(x\) sufficiently close to \(1\) which are greater than \(1\text{.}\)

Interactive Demonstration. Investigate the one sided limits \(\ds\lim_{x\to 1^+} f(x) \) and \(\ds\lim_{x\to 1^-} f(x) \) using the sliders below:

###### Definition 3.2. Left and Right-Hand Limit (Useable Definition).

In general, we write

if we can make the values of \(f(x)\) arbitrarily close to \(L\) by taking \(x\) to be sufficiently close to \(a\) and \(x\) less than \(a\text{.}\) This is called the left-hand limit of \(f(x)\) as \(x\) approaches \(a\text{.}\) Similarly, we write

if we can make the values of \(f(x)\) arbitrarily close to \(L\) by taking \(x\) to be sufficiently close to \(a\) and \(x\) greater than \(a\text{.}\) This is called the right-hand limit of \(f(x)\) as \(x\) approaches \(a\text{.}\)

We note the following fact:

###### Theorem 3.3. Connection between Limit and Left & Right Limits.

\(\ds{\lim_{x\to a}f(x)=L\qquad\mbox{if and only if} \qquad\lim_{x\to a^-}f(x)=L\qquad\mbox{and} \qquad\lim_{x\to a^+}f(x)=L}\text{.}\)

Or more concisely:

A consequence of this fact is that if the one-sided limits are *different*, then the two-sided limit \(\ds{\lim_{x\to a}f(x)}\) does not exist, often denoted as DNE.

##### Exercises for Section 3.1.

###### Exercise 3.1.1.

Use a calculator to estimate \(\ds\lim_{x\to 0} \frac{\sin x}{x}\text{,}\) where \(x\) is in radians.

Answer\(\ds\lim_{x\to 0} \frac{\sin x}{x} \approx 1\)

We can investigate the limit,

numerically (one fast way would be to use a spreadsheet). Let \(f(x) = \dfrac{\sin{x}}{x}\text{.}\)

Notice that as \(x \to 0^{+}\text{,}\) \(f(x)\) appears to be approaching \(1\text{.}\) To be complete, we need to consider the limit from the negative side,

As \(x \to 0^{-}\text{,}\) \(f(x)\) appears again to be approaching \(1\text{.}\) Hence, our estimate for \(\lim\limits_{x \to 0} \dfrac{\sin{x}}{x}\) is \(1\text{.}\)

###### Exercise 3.1.2.

Use a calculator to estimate \(\ds\lim_{x\to 0} \frac{\tan(3x)}{\tan(5x)}\text{,}\) where \(x\) is in radians.

Answer\(\ds\lim_{x\to 0} \frac{\tan(3x)}{\tan(5x)} \approx 0.6\)

First, look at the function behaviour as \(x\) approaches \(0\) from the right:

So we see that \(\lim\limits_{x \to 0^{+}} \dfrac{\tan{(3x)}}{\tan{(5x)}} \approx 0.6\text{.}\) Now look at the left-hand limit:

Therefore \(\lim\limits_{x \to 0} \dfrac{\tan{(3x)}}{\tan{(5x)}} \approx 0.6\text{.}\)

###### Exercise 3.1.3.

Use a calculator to estimate \(\ds\lim_{x\to 1^{+}}\frac{|x-1|}{1-x^2}\) and \(\ds\lim_{x\to 1^{-}}\frac{|x-1|}{1-x^2}\text{.}\)

Answer\(\ds\lim_{x\to 1^{+}}\frac{|x-1|}{1-x^2} \approx -0.5\) and \(\ds\lim_{x\to 1^{-}}\frac{|x-1|}{1-x^2} \approx 0.5\)

We first investigate \(\lim\limits_{x \to 1^{+}} \dfrac{|x-1|}{1-x^2}\text{:}\)

Therefore we estimate \(\lim\limits_{x \to 1^{+}} \dfrac{|x-1|}{1-x^2} \approx -0.5\text{.}\) Similarly, from the left-hand side we have:

and so we estimate \(\lim\limits_{x \to 1^{-}} \dfrac{|x-1|}{1-x^2} \approx 0.5\text{.}\)

In particular we see that

and so \(\lim\limits_{x \to 1} \dfrac{|x-1|}{1-x^2}\) does not exist.