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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(a)

Horizontal Asymptote

x → −∞

lim f(x) = L

or

x→ ∞

lim f(x) = L

As x decreases or increases without bound, the function approaches a numerical value. The horizontal line, y=L is an asymptote. (b) Vertical Asymptote x → c−

lim f(x) = ∞

or

x → c+

lim f(x) = ∞

As x approaches the value c (from left or right), the function increases without bound x → c−

lim f(x) = −∞

or

x → c+

lim f(x) = −∞

As x approaches the value c (from left or right), the function decreases without bound

7.

When do limits fail to exist?

B 1.

Limits – Algebraic Approach Principal Limit Theorem (Main Limit Theorem) Let c and k be real numbers, n≥0 and let f and g be functions with limits at c such that x→ c

lim f(x) = L lim k = k

or

x→ c

lim g(x) = M

(a) (b) (c)

Constant Rule: Identity Rule: Coefficient Rule:

x→ c

x→ c

lim x = c lim k f(x) = k lim f(x) = k L

x→ c

x→ c

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(d) (e) (f)

Sum Rule: Difference Rule: Product Rule:

x→ c

lim (f(x) + g(x) ) = lim f(x) + lim g(x) = L + M

x→ c x→ c x→ c x→ c

x→ c

lim (f(x) − g(x) ) = lim f(x) − lim g(x) = L − M

x→ c x→ c

x→ c

lim (f(x) . g(x) ) = lim f(x) . lim g(x) = LM

(g)

Quotient Rule:

lim f(x) f(x) x → c L lim g(x) = lim g(x) = M x→ c x→ c

lim provided that x → c g(x) ≠ 0

n = lim f(x) = Ln x→ c

(h)

Power Rule:

x→ c

lim (f(x) )

n

2.

Examples Example 1

Assume that lim f(x) = 7 and lim g(x) = −3 , find

x →b x →b

x →b

lim (f(x) + g(x) ) lim (3 f(x)g(x) ) lim 4 x g(x)

Ans : [ 4 ] Ans : [ − 63 ] Ans : [ − 12b ]

x →b

x →b

x →b

lim

g(x) f(x)

Ans : [ −

3 ] 7

Example 2 Example 3

lim (2x 3 − 3x 2 + x + 1)

x →2

Ans : [ 7 ]

Ans : [ 216 ]

x →3

lim 2x 3 x 2 + 7

Example 4

2x − 1 lim 3 x →1 x − 2

Ans : [ − 1 ]

Example...