# 11x1 t09 01 limits & continuity (2011)

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• Limits & Continuity

• Limits & ContinuityA limit describes the behaviour of functions.

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

lim :x a

f x

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

lim :x a

f xas the x value approaches a from the negative side, what value does f(x) approach?

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

lim :x a

f xas the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

lim :x a

f xas the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x

1lim 1 0x

x

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

lim :x a

f xas the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x

1lim 1 0x

x

y

x

46 y f x

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

lim :x a

f xas the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x

1lim 1 0x

x

y

x

46 y f x

0

lim 4x

f x

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

lim :x a

f xas the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x

1lim 1 0x

x

y

x

46 y f x

0

lim 4x

f x

lim :x a

f x

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

lim :x a

f xas the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x

1lim 1 0x

x

y

x

46 y f x

0

lim 4x

f x

lim :x a

f xas the x value approaches a from the positive side, what value does f(x) approach?

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

lim :x a

f xas the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x

1lim 1 0x

x

y

x

46 y f x

0

lim 4x

f x

lim :x a

f xas the x value approaches a from the positive side, what value does f(x) approach?

1lim 1 0x

x

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

lim :x a

f xas the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x

1lim 1 0x

x

y

x

46 y f x

0

lim 4x

f x

lim :x a

f xas the x value approaches a from the positive side, what value does f(x) approach?

1lim 1 0x

x 0lim 6x f x

• Limits & ContinuityA limit describes the behaviour of functions.

lim :x a

f x as the x value approaches a, what value does f(x) approach?

lim :x a

f xas the x value approaches a from the negative side, what value does f(x) approach?

y

x1

1

1y x

1lim 1 0x

x

y

x

46 y f x

0

lim 4x

f x

lim :x a

f xas the x value approaches a from the positive side, what value does f(x) approach?

1lim 1 0x

x 0lim 6x f x If lim lim , then is continuous at

x a x af x f x f x x a

• Finding Limits

• Finding Limits(1) Direct Substitution

• Finding Limits(1) Direct Substitution

5e.g. lim 7

xx

• Finding Limits(1) Direct Substitution

5e.g. lim 7

xx 5 7

12

• Finding Limits(1) Direct Substitution

5e.g. lim 7

xx 5 7

12

(2) Factorise and Cancel

• Finding Limits(1) Direct Substitution

5e.g. lim 7

xx 5 7

12

(2) Factorise and Cancel2

3

9e.g. lim3x

xx

• Finding Limits(1) Direct Substitution

5e.g. lim 7

xx 5 7

12

(2) Factorise and Cancel2

3

9e.g. lim3x

xx

33 3

lim3x

x xx

• Finding Limits(1) Direct Substitution

5e.g. lim 7

xx 5 7

12

(2) Factorise and Cancel2

3

9e.g. lim3x

xx

33 3

lim3x

x xx

3lim 3x

x

• Finding Limits(1) Direct Substitution

5e.g. lim 7

xx 5 7

12

(2) Factorise and Cancel2

3

9e.g. lim3x

xx

33 3

lim3x

x xx

3lim 3x

x 3 36

• (3) Special Limit

• (3) Special Limit1lim 0

x x

• (3) Special Limit1lim 0

x x

3 233 2 1e.g. lim 4 1xx x xi

x

• (3) Special Limit1lim 0

x x

3 233 2 1e.g. lim 4 1xx x xi

x

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

• (3) Special Limit1lim 0

x x

3 233 2 1e.g. lim 4 1xx x xi

x

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

2 3

3

3 2 11lim 14x

x x x

x

• (3) Special Limit1lim 0

x x

3 233 2 1e.g. lim 4 1xx x xi

x

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

2 3

3

3 2 11lim 14x

x x x

x

14

• (3) Special Limit1lim 0

x x

3 233 2 1e.g. lim 4 1xx x xi

x

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

2 3

3

3 2 11lim 14x

x x x

x

14

234 lim 1xx xiix

• (3) Special Limit1lim 0

x x

3 233 2 1e.g. lim 4 1xx x xi

x

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

2 3

3

3 2 11lim 14x

x x x

x

14

234 lim 1xx xiix

010

• (3) Special Limit1lim 0

x x

3 233 2 1e.g. lim 4 1xx x xi

x

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

2 3

3

3 2 11lim 14x

x x x

x

14

234 lim 1xx xiix

010

7 6 27 lim 3 974xx x xiiix x

• (3) Special Limit1lim 0

x x

3 233 2 1e.g. lim 4 1xx x xi

x

3 2

3 3 3 3

3

3 3

3 2 1

lim4 1x

x x xx x x x

xx x

2 3

3

3 2 11lim 14x

x x x

x

14

234 lim 1xx xiix

010

7 6 27 lim 3 974xx x xiiix x

13

• 32 2 lim 1xxivx

• 32 2 lim 1xxivx

10

• 32 2 lim 1xxivx

10

3 2

Find the horizontal asymptote of 1 1

x xv y

x x

• 32 2 lim 1xxivx

10

3 2

Find the horizontal asymptote of 1 1

x xv y

x x

3 2lim

1 1xx xx x

• 32 2 lim 1xxivx

10

3 2

Find the horizontal asymptote of 1 1

x xv y

x x

3 2lim

1 1xx xx x

2

26lim

1xx xx

• 32 2 lim 1xxivx

10

3 2

Find the horizontal asymptote of 1 1

x xv y

x x

3 2lim

1 1xx xx x

2

26lim

1xx xx

111

• 32 2 lim 1xxivx

10

3 2

Find the horizontal asymptote of 1 1

x xv y

x x

3 2lim

1 1xx xx x

2

26lim

1xx xx

111

horizontal asymptote is 1y

• 32 2 lim 1xxivx

10

3 2

Find the horizontal asymptote of 1 1

x xv y

x x

3 2lim

1 1xx xx x

2

26lim

1xx xx

111

horizontal asymptote is 1y Exercise 7I; 1a, 2ace, 3ac,4a, 5ad, 8a, 9ab, 10a

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