Limits and techniques for solving limits problems

Limits and continuity

By Du Tran

​To understand what limits are, let's look at an example.

We start with the function f(x)=x+2

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The limit of f at x=3 is the value f approaches as we get closer and closer to x=3. Graphically, this is the y-value we approach when we look at the graph of f and get closer and closer to the point on the graph where x=3.​

​For example, if we start at the point (1,3) and move on the graph until we get really close to x=3 then our y-value (i.e. the function's value) gets really close to 5.

​Similarly, if we start at (5,7) and move to the left until we get really close to x=3 y-value again will be really close to 5.

​For these reasons we say that the limit of f at x=3 is 5.

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​You might be asking yourselves what's the difference between the limit of f at x=3 and the value of f at x=3, i.e. f(3).

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​So yes, the limit of f(x)=x+2 at x=3 is equal to f(3), but this isn't always the case. To understand this, let's look at function g. This function is the same as f in every way except that it's undefined at x=3.

​Just like f, the limit of g at x=3 is 5. That's because we can still get very very close to x=3 and the function's values will get very very close to 5.

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So the limit of ggg at x=3 is equal to 5, but the value of g at x=3 is undefined! They are not the same!

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That's the beauty of limits: they don't depend on the actual value of the function at the limit. They describe how the function behaves when it gets close to the limit.

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Multiple Choice

What is a reasonable estimate for the limit of h at x=3?

• 2

• 3

• 4

• The limit does not exist!

​We also have a special notation to talk about limits. This is how we would write the limit of f as x approaches 3

The symbol "lim" means we're taking a limit of something.

The expression to the right of "lim" is the expression we're taking the limit of. In our case, that's the function f.

The expression x→3 that comes below "Iim" means that we take the limit of f as values of x approach 3.

Multiple Choice

What is a reasonable estimate for $$\lim_{x\to6\ }f\left(x\right)$$  

• -5

• -3

• 6

• The limit does not exist!

Multiple Choice

Which expression represents the limit of $$x^2$$  as x approaches 5?

• $$\lim\ 5^2$$  

• $$\lim_{x^2\to\ 5}$$  

• $$\lim_{x\to5}x^2$$  

• $$\lim_{x\to25}x$$  

​What do we mean when we say "infinitely close"? Let's take a look at the values of f(x)=x+2 as the x-values get very close to 3. (Remember: since we're dealing with limits we don't care about f(3) itself.)

​We can see how, when the x-values are smaller than 3 but become closer and closer to it, the values of f become closer and closer to 5.

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Notice that the closest we got to 5 was with f(2.999)=4.999 and f(3.001)=5.001 which is 0.001, units away from 5.

We can get closer than that if we want. For example, suppose we wanted to be 0.00001 units from 5, then we would pick x=3.00001 and then f(3.00001)=5.00001.

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Multiple Choice

What is a reasonable estimate for $$\lim_{x\to-7}g\left(x\right)$$  

• -7

• 6

• 6.15

• 6.33

• The limit does not exist!

A limit must be the same from both sides.

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Now take, for example, function hhh. The y-value we approach as the x-values approach x=3 depends on whether we do this from the left or from the right.

When we approach x=3 from the left, the function approaches 4. When we approach x=3 from the right, the function approaches 6.

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When a limit doesn't approach the same value from both sides, we say that the limit doesn't exist.

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Multiple Select

Which of the limit exist?

• $$\lim_{x\to3}g\left(x\right)$$  

• $$\lim_{x\to5}g\left(x\right)$$  

• $$\lim_{x\to6}g\left(x\right)$$  

• $$\lim_{x\to7}g\left(x\right)$$  

Congratulation! You have reached the end of this lecture​