What LIMIT Means Given a function: f(x) = 3x – 5 Describe its parts. The function is a line The slope of the function is 3 The y-intercept = -5 (vertical intercept) The OUTput for x = 3 is 4: f(3) = 3 · 3 – 5 = 4. Easy, everyone understands, everyone’s happy. What else does this mean, however? It means that the point (3, 4) belongs to the relation and function f. Furthermore, it means that the point (3,4 ) falls on the graph of f(x), as evidenced in this figure.

(3, 4) This is still an easy concept. But, now let’s change the way we talk just a little to prepare for limits. f (x) = 3x – 5 (0, -5)

What LIMIT Means Notice that as you get closer and closer to x = 3, the height of the graph gets closer and closer to y = 4. In fact, if you plug x = 2.9 into f(x), you get f(2.9) = 3(2.9) – 5 = 3.7 If you plug in x = 2.95, the output is 3.85. Inputs close to 3 give outputs close to 4, and the closer the input is to 3, the closer the output is to 4.

What LIMIT Means Even if you didn’t know that f(3) = 4, you could still figure out what it would probably be by plugging in an insanely close number like 2.99999. Saving us some time, f(2.99999) = 3.99997. Conclusion: It’s obvious that as f heads straight for the point (3, 4), we can state that’s what is meant by a limit.

LIMITS Define a limit: Limits are analyzed 3 ways:

LIMITS Define a limit: The limit of the function, f(x) as x approaches c is L Limits are analyzed 3 ways: Numerically (using a table) Graphically (using a graph) Algebraically (using algebra, or mathematics)

Finding limits Numerically

Use your calculator to fill in the following table for f(x): 3.9 3.99 3.999 4 4.001 4.01 4.1 y

Use your calculator to fill in the following table for f(x): 3.9 3.99 3.999 4 4.001 4.01 4.1 y .25158

Use your calculator to fill in the following table for f(x): 3.9 3.99 3.999 4 4.001 4.01 4.1 y .25158 .25016

Use your calculator to fill in the following table for f(x): 3.9 3.99 3.999 4 4.001 4.01 4.1 y .25158 .25016 .25002 undef.

Use your calculator to fill in the following table for f(x): 3.9 3.99 3.999 4 4.001 4.01 4.1 y .25158 .25016 .25002 undef. .24998 .24984 .24846 As we get closer and closer to x = 4 from both the right and left sides, does it look like y is trying to get to any particular value? What is it?

Informally, a limit is the y value that the function is trying to get to, near a certain x value. As x gets closer to 2, it looks like y is getting close to 6

This is also true even if y never actually gets to that value, when it just looks like it is going to As x gets closer to 2, y is still getting really close to 6 It doesn’t matter that f never actually gets to y = 6, just that it is getting really close.

The notation we use to describe either of the previous two situations is: This means: “The limit of f(x) as x approaches 2 is equal to 6”. It does not matter whether it actually gets to 6, just that it is getting really close.

Another informal definition If f(x) (i.e. the y value) gets closer and closer to a number L as x gets closer and closer to a number c from both sides, then L is the limit of f(x) as x approaches c. The notation below is how we write this:

Finding limits algebraically

Let’s go over the algebraic properties of limits To begin Let’s go over the algebraic properties of limits

If you have a constant function A function of the form y = k, where k is a constant (number), has a certain type of graph. What is it? So, find

If you have a constant function A function of the form y = k, where k is a constant (number), has a certain type of graph. What is it? CONSTANT So, find 5

Another basic function: y = x 1.5 Finding limits algebraically Another basic function: y = x Take a look at the graph of y = x. Find

Another basic function: y = x 1.5 Finding limits algebraically Another basic function: y = x Take a look at the graph of y = x. Find

Limits of polynomials For any polynomial, when we are analyzing limits, they are just the y-value of the corresponding x-value.

Limits of polynomials For any polynomial, when we are analyzing limits, they are just the y-value of the corresponding x-value.

Rational functions Recall, a rational function is defined as a function that is some polynomial in the numerator over some other polynomial in the denominator, as long as the denominator is not zero. Remember that the domain is a rational function is all the values of x for which the denominator is not zero.

When the denominator of a rational function IS 0, one of two things happen: At x = 1 we get a zero in the denominator, but not in the numerator. The line x = 1 is a vertical asymptote.

Case 2: Plugging in a 2 gives a zero in the numerator AND the denominator. This gives us a hole in our graph.

Understanding these differences is important when we take limits of rational functions: Is c in the domain of your function? i.e. when you plug in a c do you get a value (no zero in your denominator)? If so, then you’ve found your limit. Does c give you a zero in your denominator? If so, then you’ll need to do a little further analysis.

Example 1 Find

Example 2 Find

Example 3 Find

Example 4 Find

1.5 Finding limits algebraically Try on your own: a)Dne b) 3

LIMITS Reasons a limit does not exist (DNE) f(x) increases without bound f(x) decreases without bound f(x) oscillates between two values f(x) approaches different values from the left and right

The Definition of Limit at a Point A limit is the intended height of a function at a given value of x, whether or not the function actually reaches that height at the given x. Remember that a limit is the height a function intends to reach.

NOTATION of Limits If f(x) gets close to the number L as x approaches some value “a” from both the left (negative) and right (positive), then the limit of f(x) is L as x approaches a.

One-sided Limits As x approaches a from the RIGHT (positive-side) the limit goes to L. As x approaches a from the LEFT (negative-side) the limit goes to K. When L = K, we say that the limit exists at a and it is L.

Limits Equal to Infinity As x approaches a the limit goes off to positive infinity. As x approaches a the limit goes off to negative infinity.

What These Limits Mean When we say as x approaches a, we mean as the values of x get increasingly LARGE, the limit approaches INCREASINGLY LARGE values of f(x) and the limit acts like a vertical asymptote. Likewise, When we say as x approaches a, we mean as the values of x get increasingly LARGE, the limit approaches INCREASINGLY SMALL values of f(x) and the limit acts like a vertical asymptote.

Limits Equal to Infinity

Limits Equal to Infinity

Limits Equal to Infinity

Limit as “X” approaches Infinity As x approaches infinity the limit goes to L. As x approaches negative infinity the limit goes off to L.

What These Limits Mean When we say as x approaches negative infinity, we mean as the values of x get increasingly SMALL (moving left), the limit approaches a value L which acts like a horizontal asymptote. Likewise, When we say as x approaches infinity, we mean as the values of x get increasingly LARGE (moving right), the limit approaches a value L which acts like a horizontal asymptote.

Limit as “x” approaches Infinity

What These Limits Mean IMPORTANT NOTE: **In order for a LIMIT to exist, the left side approach must be Equal to the right side approach

Asymptotes Vertical Asymptote: any value of “x” which makes the denominator 0. Horizontal Asymptote: The numerator and the denominator have the same degree HA is   The degree of the numerator is less than the degree of the denominator: HA is y = 0 The degree of the numerator is greater the degree of the denominator: Horizontal Asymptote DOES NOT EXIST.

Limit Theorems Almost all limits are actually found by substituting the values into the expression, simplifying, and coming up with a number, the limit. The theorems on limits of sums, products, powers, etc. justify the substituting. Those that don’t simplify can often be found with more advanced theorems such as L'Hôpital's Rule

Limit Laws If the limits both exist, then:

Limit Laws If the limits both exist, then:

Limit Laws If the limits both exist, then: If f (x) ≤ g(x) ≤ h(x) & If f is continuous at b and then:

EVALUATING Limits – by Graphs

EVALUATING Limits – by Graphs

EVALUATING Limits – by Graphs

EVALUATING Limits – by Graphs

EVALUATING Limits – by Graphs

EVALUATING Limits – by Graphs

EVALUATING Limits – by Graphs

EVALUATING Limits – by Graphs

EVALUATING Limits – by Graphs

EVALUATING Limits – Algebraically

EVALUATING Limits – Algebraically SOLUTION: Recall – Almost all limits are actually found by substituting the values into the expression, simplifying, and coming up with a number, the limit.

EVALUATING Limits – Algebraically SOLUTION: Recall – Almost all limits are actually found by substituting the values into the expression, simplifying, and coming up with a number, the limit.

EVALUATING Limits – Algebraically SOLUTION: Recall – Almost all limits are actually found by substituting the values into the expression, simplifying, and coming up with a number, the limit.

Evaluating limits at infinity ∞ Let’s use the rules we just found to help us algebraically evaluate a limit at infinity. Before we start, though, enter the function on your calculator and pick large numbers for x to see what is happening to y as x gets infinitely large.

Evaluating limits at infinity ∞ The numerator and the denominator have the same degree, the limit is the ratio of the coefficients with the same variable degree

Evaluating limits at infinity ∞ Because the degree of x3 in the numerator is the same as the denominator, the limit is actually the ratio of the coefficients 1 and – 5.

Evaluating limits at infinity ∞ The degree of the numerator is less than the degree of the denominator, the limit is 0.

Evaluating limits at infinity ∞ Because the degree of x2 is in the numerator and is smaller, than the degree of the denominator, the limit is 0.

Evaluating limits at infinity ∞ The degree of the numerator is greater the degree of the denominator: Limit DOES NOT EXIST.

Evaluating limits at infinity ∞ Because the degree of x4 in the numerator, the limit is actually –∞, or Does NOT Exist.

Let’s do some more:

Let’s do some more:

Let’s do some more:

EVALUATING Limits – Algebraically

EVALUATING Limits – Algebraically SOLUTION: Recall – Almost all limits are actually found by substituting the values.

EVALUATING Limits – Algebraically

EVALUATING Limits – Algebraically SOLUTION: Recall – Almost all limits are actually found by substituting the values. Notice what happens here. Substituting does NOT always work. In SOME cases, we MUST FACTOR.

EVALUATING Limits – Algebraically SOLUTION: FACTOR first.

EVALUATING Limits – Algebraically

EVALUATING Limits – Algebraically SOLUTION: With |Absolute Value|, we must account for the Negative Values and the Positive Values. Additionally, we MUST Factor. Since the LIMITS are not equal for left nor right, then the LIMIT DNE.

CAUTION!!!!!! The way we solve limits that APPROACH infinity ∞ DOES NOT necessarily use the same methods we use to solve limits that approach a value “a” or “c”

EVALUATING Limits – Difference Quotient Find the limit for f(x) = |7 + 5x|

EVALUATING Limits – Difference Quotient SOLUTION: Evaluate the limit algebraically.

EVALUATING Limits – Difference Quotient Find the limit for using:

EVALUATING Limits – Difference Quotient SOLUTION: Evaluate the limit algebraically.

CONTINUITY Definition of Continuity: Let a be a point in the domain of the function f(x). Then f is continuous at x = a, if and only if Conditions for continuity at some real number: That number must be in the domain of the function. The function must have a limit at that number. The limit at that number must equal the value of that number in the function.

SUMMARY denotes a limit. A limit is the intended height of a function at a given value of x denotes a limit. A limit can ONLY exist when the left side = the right side Limits that are infinite occur at VERTICAL Asymptotes. The limit of a sum is the SUM of the limits The limit of a difference is the Difference of the limits The limit of a product is the Product of the limits The limit of a quotient is the Quotient of the limits Difference Quotients are used to determine limits (derivatives)