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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