AB Limits graphically class notes...One sided Limits. Notation for a one sided limit: It is the limit from the left or leftsided limit of f(x) =k whenever x is approaching from the

AB Limits graphically class notes September 01, 2015







Analysis of TablesWhat do you observe?


What do you observe?



one last onewhat do you observe?


One sided Limits Notation for a one sided limit:

It is the limit from the left or leftsided limit of f(x) =k whenever x is approaching from the left side of "c"

Similarly: a rightsided limit is denoted by


Type 1: Removable

A removable discontinuity occurs when there is a hole in the graph.

Type 2: Nonremovable

A nonremovable 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

T

There are two types of discontinuities.

"jump"vertical asymptote aka "infinite"

"Bridge is missing"


infinity


Horizontal Asymptotes

1.

2.

3.

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 existthe function is unbounded.

(bigger on top none BOTN)

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 valueEATS DC divide coefficients



Removable discontinuitiesIf the limit of a function approaches the same yvalue as x approaches "c" from the left or the right of f(x) but f(c) does not exist (the point) then a removable discontinuity is present.

#

#



Vertical AsymptotesIf 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

1.

2. If this happens from the right we say







Continuity of a functionA function is continuous at a point "c" if the following conditions are met:

2.

1.

3.

which implies the left and right limits agree

which means a point on the graph there

the point lies in the "limit"

If any of these conditions is not met, the function is "discontinuous" at the point c






When finding a limit, always use direct substitution first.

Three things could happen:1) f(x) = any # 2) 3)

AB Limits graphically class notes...One sided Limits. Notation for a one sided limit: It is the limit from the left or leftsided limit of f(x) =k whenever x is approaching from the

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