Essential Knowledge 1.1A2

Essential Knowledge 1.1A2 Students will know that the concept of a limit can be extended to include one-sided limits, limits at infinity, and infinite limits.

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Introduction to One-Sided Limits

In 1.1A1, we used two-sided limits. For , a two-sided limit checks for a limiting value using values of x which are smaller than c and values bigger than c.

Sometimes, we only want to check a limit using values of x smaller than c. These are called one-sided limits "from the left", or "left-handed limits", and are notated with a "-" superscript on c, like so:

Sometimes, we only want to check a limit using values of x larger than c. These are called one-sided limits "from the right", or "right-handed limits", and are notated with a "+" superscript on c, like so:

A regular limit (two-sided limit) only exists if the left-hand limit equals the right-hand limit.

Example 1

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.

Introduction to Limits at Infinity

Instead of a real number, limits can be taken to infinity, such as . Even though infinity is an idea and not a real number, all I ever do is just think if it as a really big number and that always works for me.

Example 2 (copied from Essential Knowledge 1.1A1)

So far limits probably seem pretty silly. Like, why not just substitute x = 5, duh? Lets do an example where the limit can be more useful than substitution.

"Given a function f,..."

Let's use f(x) = 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 something for "c" that calculus is all about: infinity (which has the symbol ∞).

Now, if you tried to solve this just using substitution, for x = ∞ would give you , which is indeterminate and therefore kinda useless.

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

You can't have a number greater than infinity so we can't make a second table.

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

The table above shows f(x) clearly getting closer and closer to 0. How close? Arbitrarily close--in other words, as close as we want. We can get closer and closer to 0 the larger we make x.

!! And 0 is a much more useful answer than indeterminate.

Therefore, this limit exists, and it is equal to 0. According to the definition of a limit, R = 0. We can say

"The limit of as x approaches ∞ equals 0."

or symbolically

The graph below also shows ln(x) / x getting closer and closer to 0 as x gets larger

Introduction to Infinite Limits

Sometimes limits don't approach a real number, but instead approach either ∞ or -∞. Since infinity isn't a real number, the limit technically doesn't exist. I personally don't like this. For me, knowing my answer gets infinitely big gives me exactly what I need to know. So I missed this question when I took a practice test a few days ago. Yeah, maybe I'm just salty. But this is just something you'll have to memorize: if your answer to a limit question is ∞ or -∞, you'll have to say the limit doesn't exist.

Example 3

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