Limits and Continuity The student will learn about: limits, limits, finding limits, one-sided limits, infinite limits, and continuity.

A limit is the intended height of a function. Simple definition: A limit is the intended height of a function.

THE LIMIT (L) OF A FUNCTION IS THE VALUE THE FUNCTION (y) APPROACHES AS THE VALUE OF (x) APPROACHES A GIVEN VALUE.

When does a limit exist? A limit exists if you travel along a function from the left side and from the right side, towards some specific value of x, as long as the function meets at the same height. A general limit exists on f(x) when x = c if the right and left hand limits are both equal

One-Sided Limit We write and call L the limit from the right (or right-hand limit) if f (x) is close to L whenever x is close to c, but to the right of c on the real number line.

One-Sided Limit We have introduced the idea of one-sided limits. We write and call K the limit from the left (or left-hand limit) if f (x) is close to K whenever x is close to c, but to the left of c on the real number line. 5

The Limit Thus we have a left-sided limit: And a right-sided limit: And in order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.

Example f (x) = |x|/x at x = 0 The left and right limits are different, therefore there is no limit.

Three methods of evaluating limits Substitution Factoring The Conjugate Method

Solving Limits using direct substitution Direct substitution is the easiest way to solve a limit Can’t use it if it gives an undefined answer

Limit Properties These rules, which may be proved from the definition of limit, can be summarized as follows. For functions composed of addition, subtraction, multiplication, division, powers, root, limits may be evaluated by direct substitution, provided that the resulting expression is defined.

Examples – FINDING LIMITS BY DIRECT SUBSTITUTION Substitute 4 for x. Substitute 6 for x.

Example 1 – FINDING A LIMIT BY Substitution and TABLES Use tables to find Solution : We make two tables, as shown below, one with x approaching 3 from the left, and the other with x approaching 3 from the right.

Limits IMPORTANT! This table shows what f (x) is doing as x approaches 3. Or we have the limit of the function as x approaches We write this procedure with the following notation. 10 Def: We write 3 or as x → c, then f (x) → L if the functional value of f (x) is close to the single real number L whenever x is close to, but not equal to, c. (on either side of c). H x 2 2.9 2.99 2.999 3 3.001 3.01 3.1 4 f (x) 8 9.8 9.98 9.998 ? 10.002 10.02 10.2 12

Limits As you have just seen the good news is that many limits can be evaluated by direct substitution.

Factoring method But be careful when a quotient is involved. But the limit exist!!!! Graph it. What happens at x = 2?

Using direct substitution is not always as evident Using direct substitution is not always as evident. Find the limit below.

Rewrite before substituting Factor and cancel common factors – then do direct substitution. The answer is 4.

The conjugate Method

Limit and Infinity  

Infinite Limits Sometimes as x approaches c, f (x) approaches infinity or negative infinity. Consider From the graph to the right you can see that the limit is ∞. To say that a limit exist means that the limit is a real number, and since ∞ and - ∞ are not real numbers means that the limit does not exist.

 

Limits at Infinity (horizontal)  

Conclusion  

Intro to Continuity As we have seen some graphs have holes in them, some have breaks and some have other irregularities. We wish to study each of these oddities. We will use our information of limits to decide if a function is continuous or has holes.

What makes a function continuous Continuous functions have: No breaks in the graph No jumps No holes

Types of discontinuity Point or hole infinite or vertical jump or gap asymptote

THIS IS THE DEFINITION OF CONTINUITY A function f is continuous at a point x = c if f (c) is defined 1. 2. 3. THIS IS THE DEFINITION OF CONTINUITY

Example f (x) = x – 1 at x = 2. f (2) = a. 1 b. The limit exist! c. Therefore the function is continuous at x = 2.

Example f (x) = (x2 – 9)/(x + 3) at x = -3 a. - 6 b. c. f (-3) = 0/0 Is undefined! b. - 6 -3 The limit exist! c. -6 Therefore the function is not continuous at x = -3. You can use table on your calculator to verify this.

Continuity Properties If two functions are continuous on the same interval, then their sum, difference, product, and quotient are continuous on the same interval except for values of x that make the denominator 0. Every polynomial function is continuous. Every rational function is continuous except where the denominator is zero.

Continuity Summary. Is the function continuity? If not what type of discontinuity is it.

Examples