Holes in a function. Even though the function does not exist at that point, the limit can still obtain that value.

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Eustacia Daniel
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1 Holes in a function For rational functions, factor both the numerator and the denominator. If they have a common factor, you can cancel the factor and a zero will exist at that x value. Even though the function does not exist at that point, the limit can still obtain that value. hole in function but the limit is still the y value of that point

2 Limit exist if the road on both sides line up regardless if the bridge exist. i.e. if the graph approaches the same value from the left or right, the limit exist.

3 One sided Limits 1.1 Limits and continuity notes plus homework night 1 Notation for a one sided limit: It is the limit from the left or left sided limit of f(x) =k whenever x is approaching from the left side of "c" Similarly: a right sided limit is denoted by

5 what about

6 TYPES OF DISCONTINUITIES There are two types of discontinuities. Type 1: Removable A removable discontinuity occurs when there is a hole in the graph. T Type 2: Non removable A non removable discontinuity occurs when there is a vertical asymptote in the graph or if you have to "jump" from one piece of the graph to another vertical asymptote aka "infinite" "jump" "Bridge is missing"

7 How to justify a "removable discontinuity"

9 What is the limit from the left,and right exist and there IS a point on the function that is the same? The function is "continuous"

11 bridge exist roads approach same spot roads and bridge meet

12 oops! use the name given or correct

13 Homework due 8/27 # 1 5, 9,17,17 22

Continuity properties of specific functions 1. 2. 3. 4. A polynomial function is continuous for all x.

) A rational function is continuous for all x except those values that cause the denominator to equal zero.

23 Continuity properties of specific functions A polynomial function is continuous for all x. (Recall that lines, quadratics, cubics etc are all special case polynomials.) A rational function is continuous for all x except those values that cause the denominator to equal zero. The nth root of a function, where n is a odd positive integer >1, is continuous for all x values. The nth root of a function, where n is an even positive integer >1, is continuous for all x values where f(x) > 0 continuous and non negative.

24 1. Vertical Asymptotes If the limit of a function fails to exists as x approaches "c" from the left because the values of f(x) are becoming very large positive (or very large negative) numbers then 2. If this happens from the right we say 3. If the behavior is the same from the left and the right we say that If any of these conditions hold, we say that x=c is a vertical asymptote of f(x)

26 "jump" discontinuity

29 What would VA's look in a table?

30 Looking at a limit zoom out and look at the big picture Looking for a function value zoom in and focus on a point

which implies the left and right limits agree 3.

31 1. Continuity of a function A function is continuous at a point "c" if the following conditions are met: which means a point on the graph there 2. which implies the left and right limits agree 3. the point lies in the "limit" If any of these conditions is not met, the function is "discontinuous" at the point c

32 When finding a limit, always use direct substitution first. Three things could happen: 1) f(x) = any # 2) 3)

Horizontal Asymptotes If the limit of a function, as x goes to positive or negative infinity approaches a single value "c", we say that a horizontal asymptote occurs at y=c.

35 Horizontal Asymptotes If the limit of a function, as x goes to positive or negative infinity approaches a single value "c", we say that a horizontal asymptote occurs at y=c. This is the same as studying the end behaviors of a function and can be determined with precalculus rules If the degree of the denominator is greater than the degree of the numerator the HA is at y=0 (bigger on bottom zero BOBO) If the degree of the numerator is greater than the degree of the denominator the HA does not exist the function is unbounded (bigger on top none BOTN) 3. If the degree of the numerator equals the degree of the denominator use the ratios of the coefficients of num to den and the horiztonal asymptote will be at that value EATS DC divide coefficients

Things you can do with Limits Limits behave

36 Things you can do with Limits Limits behave like other "creatures" in math. You can and take the root of a limit. Given that and and that L and M are real (both limits exist) you can add or subtract them you can multiply them you can divide them with care you can pull a constant through the limit 5. you can pull a limit under a radical

6.1 Polynomial Functions

6.1 Polynomial Functions Definition. A polynomial function is any function p(x) of the form p(x) = p n x n + p n 1 x n 1 + + p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers and