Limits - Substitution

As x approaches 3 from both directions, y approaches 8

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y = 2x + 2

-4

-2

0

2

4

6

8

10

12

14

16

-2 -1 0 1 2 3 4 5 6€

limx→3

2x + 2 = 8

We can find the limit by substituting x = 3 into

the equation

Practice

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limx→−2

x 2 − 9x + 2( )

Answer: The limit is 24

When we try to substitute into we get which is undefined.

If we draw the graph we find that we get a straight line with equation

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limx→1

x 2 −1x −1

€

00

€

y = x +1 but x ≠1 i.e. there is a "hole" in the graph.

The hole in the graph at x = 1 is a discontinuity. y has a value for every x except x = 1. i.e.

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f (1) does not exist

You can recognise a discontinuity because you need to lift your pen to continue

your graph. The graph below is continuous because we can draw it without having to lift the

pen.

Although , we do have a limit at x = 1.

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f (1) does not exist

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The limit is 2

Two methods to find the limit.Method 1

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Factorise x2 −1x −1

= x +1( ) x −1( )x −1

= x +1 under the condition that x ≠1

Now substitute x = 1 to get a limit of 2i.e.

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limx→1

x 2 −1x −1

= limx→1

x −1( )(x +1)x −1

= limx→1

(x +1) = 2

Method 2Use L’Hospital’s Rule

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Differentiate the top and bottom of x2 −1x −1

separately i.e. = 2x1

and then substitute x =1.

i.e. limx→1

x 2 −1x −1

= limx→1

2x1

= 2

Note: Only use this when substitution gives 0/0

Practice

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limx→−4

x + 416 − x 2

Answer: Substituting gives

Using either factorising or L’Hospital’s Rule:Limit is

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00

€

18

Discontinuities and limits

f(0.5) = 3 (Solid dot gives the value at 0.5)

But

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limx→0.5

f (x) = 2.82

Not all discontinuities have a

limit

Jump discontinuity

The graph is not heading towards the

same value so there is no limit.

Tends towards 1

Tends towards -1

Limit at x = -4 does not exist

Vertical asymptotoes: Limit does not exist.

Note: this is NOT a discontinuityf(1) =2 and

0

1

2

3

-2 -1 0 1 2 3 4

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limx→1f (x) = 2

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limx→∞

3x − 42x + 5

More Limits

Divide top and bottom by x

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limx→∞

3 − 4x

2 + 5x

= 32

When bottom power is greater than top power

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limx→∞

3x 2 − 42x 3 + 5

€

limx→∞

3x 2 − 42x 3 + 5

= 0

When top power is greater than bottom

power

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limx→∞

3x 3 − 42x 2 + 5

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limx→∞

3x 3 − 42x 2 + 5

⇒ no limit

Topic 1 Reference Page Exercise

Delta 67, 69,72, 73

6.1, 6.2,6.3, 6.5

Sidebotham 51, 58 3.1, 3.2

NuLake 5, 41 1.1, 1.11

Theta (red) 76 7.2

• identif y features of given graphs:limi ts, differentiabilit y, discontinuity,gradients, concavity

• interpret features of graph

SeniorMathematics 159 1A, 1B

Worksheet 1

What do you think the limit is at x = 0?