Engineering Technology Mathematics -Limits and Continuity-

FKB20203 Engineering Technology Mathematics 2 Lecture 2: Limits and Continuity Lecturer: Norhayati binti Bakri ( ([e-mail])

WEEK 2

Objective: To evaluate limits of a function graphically and algebraically To determine the continuity of a function at a point

Limits (a) (b) A 1. in everyday life in mathematics

Limits – Graphical Approach Examples f(x) = x + 2

 x+2 , x ≠ 2 h(x) =  , x=2  3
7 6 5 4 3 2 1 0 1 2 3 4 5 -3 -2 -1 0 1 2 3 4 5

g(x) =

x2 − 4 x −2

7 6 5 4 3 2 1 0 -3 -2 -1 0

7 6 5 4 3 2 1 0 -3 -2 -1 0 1 2 3 4 5

Finding limits:

at x= -4 at x= -3 at x= -2 at x= -1 at x= 0

at x= 1 at x= 2 at x= 3 at x= 4

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2.

One Sided Limits
(a) (b) x approaches c from the right side x approaches c from the left side oaches x → c+

lim f(x) = L

x → c−

lim f(x) = L

3.

Two Sided Limits
Two sided limits exists if and only if the one sided limits exist and are equal. x → c+

lim f(x) = lim− f(x) = L x →c
4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2

then

lim f(x) = L x→c 2.5

3

3.5

Finding one-sided limits:

at x= 1 at x= 2

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4.

Infinite Limit
As x approaches a number, the limit is infinity oaches x → c−

lim f(x) = ∞

,

lim f(x) = ∞ x→c ,

x → c+

lim f(x) = ∞

x → c−

lim f(x) = ∞

,

lim f(x) = ∞ x→c ,

x → c+

lim f(x) = ∞

Finding limits:

at x= 1 at x= 2 at x= 3 at x= 4

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5.

Limit at Infinity
As x approaches infinity (positive or negative), the limit is a numerical value x → +∞

lim f(x) = L

,

x → −∞

lim f(x) = L

Finding limits:

at x= 4 at x= -4

6.

Limits and Asymptote
Asymptotes are defined with respect to limits at infinity and infinite limits.

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