Limits - Substitution

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Limits - Substitution
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As x approaches 3 from both directions, y
approaches 8
We can find the limit by substituting x 3 into
the equation
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Practice
Answer The limit is 24
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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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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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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.
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Although , we do
have a limit at x 1.
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Two methods to find the limit.Method 1
Now substitute x 1 to get a limit of 2 i.e.
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Method 2Use LHospitals Rule
Note Only use this when substitution gives 0/0
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Practice
Answer Substituting gives Using either
factorising or LHospitals Rule Limit is
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Discontinuities and limits
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f(0.5) 3 (Solid dot gives the value at 0.5)
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But
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Not all discontinuities have a limit
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Jump discontinuity
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The graph is not heading towards the same value
so there is no limit.
Tends towards 1
Tends towards -1
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Limit at x -4 does not exist
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Vertical asymptotoes Limit does not exist.
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Note this is NOT a discontinuityf(1) 2 and
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More Limits
Divide top and bottom by x
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When bottom power is greater than top power
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When top power is greater than bottom power
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Worksheet 1
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What do you think the limit is at x 0?