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Math 250 Brigs 2.2 Definitions of Limits
Objectives:
1) Use limit notation /(x), lim/(x) = L x-+a
a. Note that limf(x) = L is a limit of a single function, as opposed to the limit of an expression, X-+a
like the slope of the secant line, Jim I (x )-I (a) x-+a x-a
b. L is a finite, real number on the y-axis
c. a is an x-coordinate
d. As the x-coordinates get dose to a, they-coordinates get dose to L.
2) Estimate limits using graphs. 3) Estimate limits using tables. 4) Use one-sided limit notation and concepts correctly
a. Right-side limit: lim /(x) x-+a•
b. Left-side limit lim f (x) x-+a-
c. Relationship of a limit lim/(x) (two-sided) to the one-sided limits lim /(x) and lim /(x). x-+a x-+a+ x-+a ...
lim/(x)= lim f(x)= lim f(x) x-+a x-+a+ i-+a-
d. If the limit from the right does not equal the limit from the left, we say the limit (t.Wo-sided)
does not exist.
e. A limit with a .+or - sign in the limit notation is always a one-sided limit. f. Limits without a+ or - sign in the limit notation are two-sided, unless x approaches infinity.
("Limits at infinity", as x approaches infinity, are in a later section.) 5) Find limits of graphs with holes
a. Recognize errors of graphs with holes when graphed on the graphing calculator
b. Find limits when the function is not defined at a hole. c. Find limits when the function is defined piecewise at the hole.
d. Notice that limf(x) is not necessarily equal to f(a). X-+a
6) Three reasons that limits do not exist: a. L :1: R: The Umit of the function from the left does not equal the limit of the function from the
right: lim f (x) * lim f (x) x-.a• x-t-a-
b. Oscillation: A specific bizarre behavior near the point x = a where the function has many different y-coordinates as the x-coordinates get close to a
c. Unbounded: (L is not finite -- section 2.4)
7) Use the greatest integer function, also called the floor function, l x J. a. Evaluate b. Graph c. Find limits.
Examples
For each limit, a) Graph the function and note whether the graph can be found accurately using a graphing calculator. b) Estimate liin f (x) and liin f (x) using the graph and/or a table of values near x =a.
x-HJ• x-+a-
C) Use results to estimate liin/(x).
d) Find /(a).
e) Is liin f(x) = f(a)? x-+a
1) liin(x2 + 2) x-+3
2) lnn{<x2 + 2) x * 3 X-+3 undefined x = 3
3) 1nn{<x2 + 2) x * 3 x-+3 16 x=3
~ 4) lnn{<x2 + 2) x < 3 x-+3 x-4 x>3
5) lim col.!) X-+O \x
6) liin x-l X-+l £-1
8) limr~-~J X-+4 X-4
~ 9) IimlxJ X-+l
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