Introducing Limits

Approaching Values

Sometimes it's not possible to work out something exactly, but it is possible to work out what value it should be from its values at points that are nearby. The idea of working out what something approaches is called "finding a limit".

Here's an example of a limit:

Two (particularly stupid) cows are 36 metres apart, and walk towards each other on a collision course at a rate of 18 metres per hour. At the same time, a suicidal fly takes off from the nose of one cow and flies towards the other cow at a rate of 36 metres per hour. As soon as the fly reaches the second cow's nose, it instantaneously turns around and heads back towards the other cow at a rate of 36 metres per hour. It keeps doing this until the cows collide, squashing the fly between them and killing it. A maths teacher, who has nothing better to do, wonders what distance the fly covers before the collision and asks you to work it out.

You might start to solve this problem as follows:

The speed of the fly is twice the speed of the cows, so the fly's first lap is \( {2\over3} (36) = 24m \) long.

The second lap has length \( {1\over3} (24) = 8m \), and the length of the third lap is \( {1\over3} (8) = {8\over3}m \)

If you continue on like this, to find the total distance travelled by the fly, you need to add up

\( 24 + 8 + {8\over3} + {8\over9} + ... \)

As this sum goes on forever, we can't simply add the terms to work it out, but we can guess its value by looking at a table of values:

From the table, it looks like the total distance travelled by the fly is getting closer and closer to \( 36 m \). However, you'd think the fly would get sick of this game, give up, and fly away before an hour was up and the cows had collided.

Let's look at another problem of the sort you might come up against in a maths lesson.

Example

What is the value of \( (x^2 - 4)\over(x+2)\) at \( x = -2 \)

Let's try and plug \( x = -2 \) into the equation:

\( {(x^2 - 4)\over(x+2)} = {(-2^2 - 4)\over(-2+2)} = {(4 - 4)\over(-2+2)} = {(0)\over(0)}\)

which is BAD because \( 0\over0 \) is undefined (you might see \( 0\over0 \) called an indeterminate in some books).

So we can't find the value of \( (x^2 - 4)\over(x+2)\) at \( x = -2 \). However, we can ask a related question:

What value does \( (x^2 - 4)\over(x+2)\) approach as \( x \) gets closer and closer to \( -4 \) ?

We'd really like to say that the value of \( (x^2 - 4)\over(x+2)\) is equal to \( -4 \), when \( x = -2 \) but we can't because we get an indeterminate form when we plug in \( x = -2 \)

Mathematicians have a special word to describe what's going on in this situation: they call

\( -4 \) the limit of \( (x^2 - 4)\over(x+2)\) as \( x \) approaches \( -2 \), and write this in symbols as
\( \lim_{x \to -2} {(x^2 - 4)\over(x+2)} = -4 \)

The graph of \( (x^2 - 4)\over(x+2)\) looks like this

Did you notice: Do you notice the hole at \( x = -2 \)? Mathematicians call holes like this removable discontinuities – more about this later. You can see from the graph that, as the \( x-values \) get closer to \( -2 \), the values of {(x^2 - 4)\over(x+2)} = -4 \) get closer to \( -4 \).

We need to check both sides of a limit!

The last example was a bit like starting to walk across a bridge, and finding there was a hole in the middle. If it was just a little hole, you could jump across it and keep going, but what would happen if the other half of the bridge was missing, or if it was out of line with the side you were standing on? If you couldn't turn around in time, you might get very wet!

Limits are just like this bridge. We need to make sure that the two halves are in line, so we need to test the limit from both directions (above and below the number we're approaching) to make sure it is defined.

Example (continued):

Let's test our "limit" from the other side by checking some values of \( x \) that are smaller than \( -2 \):

As these values are definitely getting closer and closer to \( -4 \), it looks like this is the limit after all.

What happens if you check both sides and get different values?

This might happen for a function \( f(x) \) like the one in the picture below, which has a break in the middle.

If we look at what happens as \( x \) gets closer to \( 4 \) from "below" (i.e. the left), \( f(x) \) gets closer and closer to \( 1.7 \). Mathematicians would write

\( \lim_{x \to 4-} f(x) = 1.7 \)

What about as \( x \) gets closer to \( 4 \) from "above" (i.e. the right)? \( f(x) \) approaches \( 0.8 \). Mathematicians would write

\( \lim_{x \to 4+} f(x) = 0.8 \)

BUT these two values don't match up! So the limit of \( f(x) \) as \( x \) approaches \( 4 \) does not exist.

Making it more formal (a.k.a keeping the mathematicians happy!)

Start with a function \( f(x) \) and a number \( a \). The left-hand limit of \( f(x) \) as \( x \) approaches \( a \) is the number \( l \) that \( f(x) \) tends towards as \( x \) gets closer to \( a \) from below.

In symbols: $$ \lim_{x \to a-} f(x) = l $$

The left-hand limit of a function ALWAYS exists

The right-hand limit of \( f(x) \) as \( a \) approaches \( a \) is the number \( m \) that \( f(x) \) approaches as \( x \) gets closer to \( a \) from above.

In symbols: $$ \lim_{x \to a+} f(x) = m $$

The right-hand limit of a function ALWAYS exists. Note that we use a plus sign (+) under the limit to indicate that \( x \) is approaching \( a \) from the right (or above).

Finally, the limit as \( x \) approaches \( a \) of \( f(x) \) exists exactly when the left-hand and right-hand limits at \( a \) are equal:

In symbols: $$ \lim_{x \to a+} f(x) = \lim_{x \to a-} f(x) $$

If they are not equal, then we say that the limit of \( f(x) \) as \( x \) approaches \( a \) does not exist.

Do we only find limits of problem functions?

No! We can find limits even when the function value is defined at a point. They are actually used to formally define continuity for functions, something we'll talk about later.

Example

Find \( \lim_{x \to 7} {(3x + 1)\over2} \)

This function is defined at \( x = 7 \) and it is continuous there too (you don't lift your pencil when you're drawing its graph). So we can just plug \( x = 7 \) into the function definition to find

\( \lim_{x \to 7} {(3x + 1)\over2} = {(3(7) + 1)\over2} = 11\)

If you have nothing more interesting to do, you can check this by sitting and testing values of \( x \) that get closer and closer to \( 7 \) (from above and below)!

Limits as \( x \) gets really, really large

Mathematicians call these limits as \( x \) approaches infinity. Contrary to popular belief, infinity (\( \infty \) ) isn't a number, it's an idea! It's used to describe something that never ends – the numbers get larger and larger without stopping. So we can talk about limits as \( x \) approaches infinity, but we can't evaluate functions at \( \infty \).

Let's have a look at an example:

What happens to \( f(x) = {3\over x} \) as \( x \) gets very large?

We can't calculate \( f(\infty) = {3\over\infty} \) because it is undefined. However, we can try larger and larger values of \( x \) and see what happens to \( f( \) as \( x \) approaches \( \infty \):

It looks as if \( 3 \over x \) gets closer to \( 0 \) as \( x \) approaches \( \infty \). We can confirm this by looking at the graph of \( f(x) \):

The graph is getting closer and closer to the \( x-axis \) as the \( x-values \) get larger and larger

We say that

the limit as \( x \) tends to \( \infty \) of \( 3 \over x \) is \( 0 \) ,

and write

$$ \lim_{x \to \infty} {3 \over x} = 0 $$

If you want another example of a limit as \( x \) tends to \( \infty \) , have a think about the cow example that we started out with.

Finding these limits

So far we've just worked out what the limits are by plugging in values of \( x \) and deciding what it "looks like" the function is doing. Unfortunately, this isn't really good enough – strange things could occur at the values that we haven't tested, and we can never test all the possible values because there are infinitely many of them.

There are many different techniques that can be used to find limits. We'll discuss some of them in later articles.