Idea of Limits

Idea of Limits

• Think of a simple function like X plus three.

• The concept of a limit is just this.

• If we plug in numbers that are very close to 1 We’ll get back numbers that are very close to 4.

• To express this limit we write I am for limit as x approaches one of our function X plus three is equal to 4.

• In general we say that the limit as x approaches a number a of a function f of x is equal to l n l is the limit of the function.

• Here’s another example.

• Think of the function the quantity x minus two times the quantity x plus two all divided by the quantity x minus two.

• Because you can cancel an X minus two from the numerator and denominator the function will simplify to the graph.

• The line X plus 2.

• However because setting X equal to 2 in the original function will cause the denominator to be equal to zero.

• This function is undefined at that point.

• Therefore we have to modify our graph of the line X plus two to show that the function is undefined when X is two.

• So we draw an empty circle to indicate that we don’t know the function’s value.

• Keep in mind that this makes the function discontinuous at that point which we’ll talk about more later on in this video.

• If the function is undefined there then what can we say about the value at that point.

• Well we can’t give the actual value but we can give the limit if we look at the graph we can see the value of the function gets close to 4.

• As you approach X equals two from either side.
• Therefore the limit as x approaches 2 of this function is equal to 4.

Solving For Limits

Substitution

• When it comes to solving for limits.
• Notice that in both of the previous examples we could have just plug in the number we’re approaching and found the limit of the function that’s called substitution or just get all plug and chug limit problems
• don’t usually work out that easily though more often than not if we try substitution we’ll end up with either zero over zero or zero in just the denominator of our function both of which are obviously undefined .
• In those cases we’ll have to use other techniques to solve for the limit.

Factoring

• Try factoring the numerator and denominator as much as possible and then cancelling terms if your function contains a fraction and a non fraction.

Common Denominator

• Try finding a common denominator and then simplifying as much as possible.

Conjugate

• If you have a radical or square root sign Try multiplying by the conjugate.

• No matter what you use your goal will be to simplify the function enough so that you can eventually get back to substitution.

• If none of these work will have to get a little fancier but will say those methods are another note .

Infinite Limits

• Sometimes you have to find the limit not as x approaches a number a but as x approaches positive or negative infinity.

• If you’re asked to find an infinite limit you can use the following three rules as shortcuts

• if the degree of the numerator and denominator are equal to one another.

• Use ratio of coefficients to find the limit.

• If the degree of the numerator is less than the degree of the denominator then

• the infinite limit will be zero regardless of whether you approach negative or positive infinity.

• If the degree of the numerator is greater than the degree of the denominator

• then the infinite limit will be positive infinity.

• If you’re approaching positive infinity and negative infinity.

• If you’re approaching negative infinity.

Continuity

• Now that we understand the basics of limits let’s take a couple of minutes to talk about continuity .

• A function is continuous.

• If there are no holes breaks jumps or gaps of any kind and its graph you can also think about it this way.

• A function is continuous.

• If you can draw the entire thing without picking up your pen or pencil off the page if at any point on the graph you have to pick your pencil above the page and jump to the next section to continue drawing The function is discontinuous there.

Discontinuities

• While we’re at it let’s take some time to classify the most common types of discontinuities.

• Most notably point jump and infinite discontinuity

• point does continuities look like this.

• They represent specific points on the graph of a function where the function is not continuous.

• Point is continuities are an example of what we call a removable discontinuity.

• They are immovable because we can write a simple equation that will find the function at the point of just continuity and the second equation will plug the hole in the graph.

• The fact that we can plug the hole makes the discontinuity or movable .

• jump discontinuities are non-removable discontinuities.

• Because you can’t write a simple equation that fills in the gap jumped just continuities or breaks in the graph that look like this.

• Think about walking along on the graph of the function and having to jump across the discontinuity to the next section of the graph

• infinite discontinuities are discontinuities in the graph at asymptotes

• whether vertical horizontal or slant infinite discontinuities are also non-removable because you can’t write a simple function that will fill the gap and make the function continuous.

Crazy Graphs

Let’s bring this all together and use what we know so far about limits in continuity to identify what’s happening at each of these five points on this crazy graph function.

At the first point we have a point of continuity at X equals negative 6.

The value of the function there is undefined but the limited function of x approaches negative 6 is negative 2.

Whether you approach negative six from the left side or the right side the function’s value is approaching negative 2.

At the second point we have an infinite discontinuity at X equals negative 3.

The value of the function there is undefined.

Technically the limit is also undefined but sometimes we like to say that the limit is positive infinity

.

Since the graph approaches positive infinity from both sides of negative 3.

If the graph of the function approached positive infinity from one side of the asymptote and negative

infinity from the other then the limit would be truly undefined

at the third point. The graph as another point is continuity. X equals negative 1.

The value of the function there is negative 3 because of the colored circle being shown there.

But the limit of the function is negative too.

Because as you trace the graph with your finger from both the left and right side the value you’re approaching

is negative too even though the function’s value is defined as negative 3.

The fourth point we can see that the graph has a jump is continuity at X equals 1 jumped continuity

continuities require that the graph has different one sided limits.

Up to now we’ve only talked about the general limit but a function can have different left and right

hand limits a general limit only exists if the left and right hand limits are equal which will never

be the case at a jump discontinuity at X equals 1.

The left hand limit or the limit as we approach 1 from the negative side is equal to to the right hand

limit as we approach from the positive side is equal to 3.

Because these limits aren’t equal.

There is no general limit.

The value of the function is equal to 3.

Since that is the shaded point

• At the fifth and final point.
• The graph has another jump discontinuity with a left hand limit of 3.
• A right hand limit of 1.
• And since these aren’t equal no general limit the value of the function here is 2.
• Since that is a shaded point