Essential Knowledge 1.1A3

Essential Knowledge 1.1A3 Students will know that a limit might not exist for some functions at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right.

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There are 3 main cases where a limit is said to not exist. We have already seen examples of all three, so I'll just copy paste them to here.

Function is unbounded

This just means that the answer to the limit is ∞ or -∞

Example 1

This is the limit where x just gets smaller and smaller. As usual, let's try to solve this a few different ways.

Direct Substitution

Direct substitution is always worth a try. Well, what happens if I just substitute -∞ in for x?

= (-∞)² + 1

Even though infinity times infinity feels like it should be mega super infinity, remember infinity represents an idea, not a number. Infinity can't get any bigger, so (-∞)² = ∞. Using the same logic, ∞ + 1 = ∞.

So hey, looks like our answer is ∞. Let's check with a table and a graph now to be sure.

Table

We can't approach -∞ from the left, because nothing can be smaller than -∞

From the right, we basically just have to make the most negative numbers possible

As you can see, the further left we go (in other words x → -∞), the larger the value of f(x) gets (in other words, f(x) → ∞)

All three methods give us the same conclusion, and so we can write

Remember, even though an answer of ∞ gives us very useful information, the limit officially "does not exist"

Function is Oscillating Near This Value

This usually involves a trig function, since those are the guys that oscillate. And it usually involves trig functions using infinity, since that would make them oscillate really fast and thus not have a clear value.

Example 2

Finally, let's look at a limit that doesn't exist

"Given a function f,..."

Let's use for our function.

"...the limit of f(x) as x approaches c... by taking x sufficiently close to c (but not equal to c)."

Let's use the other number that calculus is all about, 0.

Now, if you tried to solve this just using substitution, for x = 0 would give you , which is undefined (because anything divided by 0 is undefined) and therefore kinda useless.

Using limits, this means we'll use numbers for x that just keep getting closer and closer to 0.

We also need to approach 0 using numbers greater than 0.

"...is a real number R if f(x) can be made arbitrarily close to R..."

The tables above do not show f(x) clearly getting closer and closer to any number. In fact, looking at either table, values of f(x) are all over the place, getting bigger, smaller, positive, negative. This makes sense, because we know sin(x) just keeps oscillating between -1 and 1 as x increases, so it will never actually settle on any number.

Therefore, this limit does not exist. We can say

"The limit of as x approaches 0 does not exist

or symbolically

The graph below shows how, as x approaches 0, the graph just starts oscillating up and down like crazy, so the limit could never hope to settle on an actual number

Limit From the Left does not Equal the Limit From the Right

Remember, for a limit to exist, the limit from the left (aka left-handed limit aka limit using numbers smaller than the target we're approaching) has to equal the limit from the right (aka right-handed limit aka limit using numbers larger than the target we're approaching)

Example 3

Determine the following limits.

a)

Substitution is always worth a try, let's see here, , nope not looking good.

Lets try a table. Since this is a left-handed limit, I only need to check values smaller than 0

Well that was easy, looks like the left-handed limit is definitely -1. Let's graph from the left of 0 to see if this seems true.

The graph also shows the value of f(x) = -1 as x approaches 0 from the left. Therefore, we can say the limit of as x approaches 0 from the left is -1, or symbolically.

b)

Let's use another table

Looks like this time the limit is 1. Let's graph from the right of 0 to check.

The graph also shows the value of f(x) = 1 as x approaches 0 from the right. Therefore, we can say the limit of as x approaches 0 from the right is 1, or symbolically.

c)

Based on the results from above, this limit cannot exist because the left-hand limit is different from the right-hand limit. The graph of also shows this.

As you can see, doesn't approach a single number as x approaches 0, so the limit doesn't exist.