11 x1 t09 01 limits & continuity (2012)

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<ul><li><p>Limits &amp; Continuity</p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x</p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p><p> lim :x a</p><p>f x</p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p><p> lim :x a</p><p>f xas the x value approaches a from the negative side, what value does f(x) approach? </p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p><p> lim :x a</p><p>f xas the x value approaches a from the negative side, what value does f(x) approach? </p><p>y</p><p>x1</p><p>1</p><p>1y x </p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p><p> lim :x a</p><p>f xas the x value approaches a from the negative side, what value does f(x) approach? </p><p>y</p><p>x1</p><p>1</p><p>1y x </p><p>1lim 1 0x</p><p>x </p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p><p> lim :x a</p><p>f xas the x value approaches a from the negative side, what value does f(x) approach? </p><p>y</p><p>x1</p><p>1</p><p>1y x </p><p>1lim 1 0x</p><p>x </p><p>y</p><p>x</p><p>46 y f x</p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p><p> lim :x a</p><p>f xas the x value approaches a from the negative side, what value does f(x) approach? </p><p>y</p><p>x1</p><p>1</p><p>1y x </p><p>1lim 1 0x</p><p>x </p><p>y</p><p>x</p><p>46 y f x</p><p> 0</p><p>lim 4x</p><p>f x </p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p><p> lim :x a</p><p>f xas the x value approaches a from the negative side, what value does f(x) approach? </p><p>y</p><p>x1</p><p>1</p><p>1y x </p><p>1lim 1 0x</p><p>x </p><p>y</p><p>x</p><p>46 y f x</p><p> 0</p><p>lim 4x</p><p>f x </p><p> lim :x a</p><p>f x</p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p><p> lim :x a</p><p>f xas the x value approaches a from the negative side, what value does f(x) approach? </p><p>y</p><p>x1</p><p>1</p><p>1y x </p><p>1lim 1 0x</p><p>x </p><p>y</p><p>x</p><p>46 y f x</p><p> 0</p><p>lim 4x</p><p>f x </p><p> lim :x a</p><p>f xas the x value approaches a from the positive side, what value does f(x) approach? </p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p><p> lim :x a</p><p>f xas the x value approaches a from the negative side, what value does f(x) approach? </p><p>y</p><p>x1</p><p>1</p><p>1y x </p><p>1lim 1 0x</p><p>x </p><p>y</p><p>x</p><p>46 y f x</p><p> 0</p><p>lim 4x</p><p>f x </p><p> lim :x a</p><p>f xas the x value approaches a from the positive side, what value does f(x) approach? </p><p>1lim 1 0x</p><p>x </p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p><p> lim :x a</p><p>f xas the x value approaches a from the negative side, what value does f(x) approach? </p><p>y</p><p>x1</p><p>1</p><p>1y x </p><p>1lim 1 0x</p><p>x </p><p>y</p><p>x</p><p>46 y f x</p><p> 0</p><p>lim 4x</p><p>f x </p><p> lim :x a</p><p>f xas the x value approaches a from the positive side, what value does f(x) approach? </p><p>1lim 1 0x</p><p>x 0lim 6x f x </p></li><li><p>Limits &amp; ContinuityA limit describes the behaviour of functions.</p><p> lim :x a</p><p>f x as the x value approaches a, what value does f(x) approach? </p><p> lim :x a</p><p>f xas the x value approaches a from the negative side, what value does f(x) approach? </p><p>y</p><p>x1</p><p>1</p><p>1y x </p><p>1lim 1 0x</p><p>x </p><p>y</p><p>x</p><p>46 y f x</p><p> 0</p><p>lim 4x</p><p>f x </p><p> lim :x a</p><p>f xas the x value approaches a from the positive side, what value does f(x) approach? </p><p>1lim 1 0x</p><p>x 0lim 6x f x If lim lim , then is continuous at </p><p>x a x af x f x f x x a </p></li><li><p>Finding Limits</p></li><li><p>Finding Limits(1) Direct Substitution</p></li><li><p>Finding Limits(1) Direct Substitution</p><p>5e.g. lim 7</p><p>xx </p></li><li><p>Finding Limits(1) Direct Substitution</p><p>5e.g. lim 7</p><p>xx 5 7</p><p>12 </p></li><li><p>Finding Limits(1) Direct Substitution</p><p>5e.g. lim 7</p><p>xx 5 7</p><p>12 </p><p>(2) Factorise and Cancel</p></li><li><p>Finding Limits(1) Direct Substitution</p><p>5e.g. lim 7</p><p>xx 5 7</p><p>12 </p><p>(2) Factorise and Cancel2</p><p>3</p><p>9e.g. lim3x</p><p>xx</p></li><li><p>Finding Limits(1) Direct Substitution</p><p>5e.g. lim 7</p><p>xx 5 7</p><p>12 </p><p>(2) Factorise and Cancel2</p><p>3</p><p>9e.g. lim3x</p><p>xx</p><p> 33 3</p><p>lim3x</p><p>x xx</p></li><li><p>Finding Limits(1) Direct Substitution</p><p>5e.g. lim 7</p><p>xx 5 7</p><p>12 </p><p>(2) Factorise and Cancel2</p><p>3</p><p>9e.g. lim3x</p><p>xx</p><p> 33 3</p><p>lim3x</p><p>x xx</p><p>3lim 3x</p><p>x </p></li><li><p>Finding Limits(1) Direct Substitution</p><p>5e.g. lim 7</p><p>xx 5 7</p><p>12 </p><p>(2) Factorise and Cancel2</p><p>3</p><p>9e.g. lim3x</p><p>xx</p><p> 33 3</p><p>lim3x</p><p>x xx</p><p>3lim 3x</p><p>x 3 36</p></li><li><p>(3) Special Limit</p></li><li><p>(3) Special Limit1lim 0</p><p>x x</p></li><li><p>(3) Special Limit1lim 0</p><p>x x</p><p> 3 233 2 1e.g. lim 4 1xx x xi</p><p>x </p></li><li><p>(3) Special Limit1lim 0</p><p>x x</p><p> 3 233 2 1e.g. lim 4 1xx x xi</p><p>x </p><p>3 2</p><p>3 3 3 3</p><p>3</p><p>3 3</p><p>3 2 1</p><p>lim4 1x</p><p>x x xx x x x</p><p>xx x</p></li><li><p>(3) Special Limit1lim 0</p><p>x x</p><p> 3 233 2 1e.g. lim 4 1xx x xi</p><p>x </p><p>3 2</p><p>3 3 3 3</p><p>3</p><p>3 3</p><p>3 2 1</p><p>lim4 1x</p><p>x x xx x x x</p><p>xx x</p><p>2 3</p><p>3</p><p>3 2 11lim 14x</p><p>x x x</p><p>x</p></li><li><p>(3) Special Limit1lim 0</p><p>x x</p><p> 3 233 2 1e.g. lim 4 1xx x xi</p><p>x </p><p>3 2</p><p>3 3 3 3</p><p>3</p><p>3 3</p><p>3 2 1</p><p>lim4 1x</p><p>x x xx x x x</p><p>xx x</p><p>2 3</p><p>3</p><p>3 2 11lim 14x</p><p>x x x</p><p>x</p><p>14</p></li><li><p>(3) Special Limit1lim 0</p><p>x x</p><p> 3 233 2 1e.g. lim 4 1xx x xi</p><p>x </p><p>3 2</p><p>3 3 3 3</p><p>3</p><p>3 3</p><p>3 2 1</p><p>lim4 1x</p><p>x x xx x x x</p><p>xx x</p><p>2 3</p><p>3</p><p>3 2 11lim 14x</p><p>x x x</p><p>x</p><p>14</p><p> 234 lim 1xx xiix</p></li><li><p>(3) Special Limit1lim 0</p><p>x x</p><p> 3 233 2 1e.g. lim 4 1xx x xi</p><p>x </p><p>3 2</p><p>3 3 3 3</p><p>3</p><p>3 3</p><p>3 2 1</p><p>lim4 1x</p><p>x x xx x x x</p><p>xx x</p><p>2 3</p><p>3</p><p>3 2 11lim 14x</p><p>x x x</p><p>x</p><p>14</p><p> 234 lim 1xx xiix</p><p>010</p></li><li><p>(3) Special Limit1lim 0</p><p>x x</p><p> 3 233 2 1e.g. lim 4 1xx x xi</p><p>x </p><p>3 2</p><p>3 3 3 3</p><p>3</p><p>3 3</p><p>3 2 1</p><p>lim4 1x</p><p>x x xx x x x</p><p>xx x</p><p>2 3</p><p>3</p><p>3 2 11lim 14x</p><p>x x x</p><p>x</p><p>14</p><p> 234 lim 1xx xiix</p><p>010</p><p> 7 6 27 lim 3 974xx x xiiix x </p></li><li><p>(3) Special Limit1lim 0</p><p>x x</p><p> 3 233 2 1e.g. lim 4 1xx x xi</p><p>x </p><p>3 2</p><p>3 3 3 3</p><p>3</p><p>3 3</p><p>3 2 1</p><p>lim4 1x</p><p>x x xx x x x</p><p>xx x</p><p>2 3</p><p>3</p><p>3 2 11lim 14x</p><p>x x x</p><p>x</p><p>14</p><p> 234 lim 1xx xiix</p><p>010</p><p> 7 6 27 lim 3 974xx x xiiix x </p><p>13</p></li><li><p> 32 2 lim 1xxivx</p></li><li><p> 32 2 lim 1xxivx</p><p>10</p></li><li><p> 32 2 lim 1xxivx</p><p>10</p><p> 3 2</p><p> Find the horizontal asymptote of 1 1</p><p>x xv y</p><p>x x </p></li><li><p> 32 2 lim 1xxivx</p><p>10</p><p> 3 2</p><p> Find the horizontal asymptote of 1 1</p><p>x xv y</p><p>x x </p><p>3 2lim</p><p>1 1xx xx x </p></li><li><p> 32 2 lim 1xxivx</p><p>10</p><p> 3 2</p><p> Find the horizontal asymptote of 1 1</p><p>x xv y</p><p>x x </p><p>3 2lim</p><p>1 1xx xx x </p><p>2</p><p>26lim</p><p>1xx xx </p></li><li><p> 32 2 lim 1xxivx</p><p>10</p><p> 3 2</p><p> Find the horizontal asymptote of 1 1</p><p>x xv y</p><p>x x </p><p>3 2lim</p><p>1 1xx xx x </p><p>2</p><p>26lim</p><p>1xx xx </p><p>111</p></li><li><p> 32 2 lim 1xxivx</p><p>10</p><p> 3 2</p><p> Find the horizontal asymptote of 1 1</p><p>x xv y</p><p>x x </p><p>3 2lim</p><p>1 1xx xx x </p><p>2</p><p>26lim</p><p>1xx xx </p><p>111</p><p>horizontal asymptote is 1y </p></li><li><p> 32 2 lim 1xxivx</p><p>10</p><p> 3 2</p><p> Find the horizontal asymptote of 1 1</p><p>x xv y</p><p>x x </p><p>3 2lim</p><p>1 1xx xx x </p><p>2</p><p>26lim</p><p>1xx xx </p><p>111</p><p>horizontal asymptote is 1y Exercise 7I; 1a, 2ace, 3ac,4a, 5ad, 8a, 9ab, 10a </p><p>Slide Number 1Slide Number 2Slide Number 3Slide Number 4Slide Number 5Slide Number 6Slide Number 7Slide Number 8Slide Number 9Slide Number 10Slide Number 11Slide Number 12Slide Number 13Slide Number 14Slide Number 15Slide Number 16Slide Number 17Slide Number 18Slide Number 19Slide Number 20Slide Number 21Slide Number 22Slide Number 23Slide Number 24Slide Number 25Slide Number 26Slide Number 27Slide Number 28Slide Number 29Slide Number 30Slide Number 31Slide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Slide Number 38Slide Number 39Slide Number 40Slide Number 41Slide Number 42</p></li></ul>