Limits

Limit of a function

If a function is defined by clip_image002, then the values of clip_image004 depend on the values of clip_image006.

As the values of clip_image008 are getting closer and closer to clip_image010also gets arbitrarily close to a single value clip_image012, say clip_image014.

Mathematically this limit is denoted by:

clip_image016

This is read as “the limit of clip_image018 as clip_image006 approaches clip_image020 , is clip_image022”. This is illustrated diagrammatically in fig 1.0 below:

clip_image023clip_image025

Finding the limit of a Polynomial function

If  clip_image027  is a polynomial function and clip_image020 is any real number, then the limit ofclip_image029 as clip_image006 approaches clip_image031 is the same as the function value at ‘clip_image033’. That is:

clip_image035

Example 1

Evaluate clip_image037

Solution

clip_image039 is a polynomial function.

clip_image041

clip_image043

clip_image045

clip_image047

Example 2

Evaluate clip_image049

Solution

clip_image051 is also a polynomial.

clip_image053

clip_image055

clip_image057

clip_image059

Example 3

Evaluate clip_image061

Solution

clip_image063

clip_image065

clip_image067

clip_image069

Example 4

Use the graph of y= figure 1.1 to evaluate:

a) clip_image071

b) clip_image073

c) clip_image075

clip_image077

 

Solution

clip_image079

a) clip_image081

b) clip_image083

c) clip_image085

Finding the limit of a rational function

If clip_image027 is a rational function given by clip_image087, for any real number r,

clip_image089

where clip_image091.

NB: If clip_image093 yields clip_image095, then we must further simplify clip_image027 before we evaluate.

Example 5

Evaluate clip_image097

Solution

Since the denominator is not zero when clip_image099, we can substitute clip_image099 directly into the function:

clip_image101

clip_image103

clip_image105

clip_image107

Example 6

Evaluate clip_image109

Solution

Again the denominator is not zero when clip_image111, we can substitute clip_image111 directly into the function:

clip_image113

clip_image115

clip_image117

Example 7

Evaluate clip_image119

Solution

Direct substitution yields, clip_image121. This is an implication that we need to simplify the given rational function before we evaluate.

clip_image123

clip_image125

clip_image127

clip_image129

Example 8:

Evaluate

clip_image131

 

Solution

Again direct substitution yields, clip_image133. Therefore we need to simplify the given rational function before we evaluate.

Recall from binomial expansion that clip_image135. We can rearrange to get an expression for the numerator, clip_image137.

That is:

clip_image139

clip_image141

clip_image143

Hence,

clip_image145

clip_image147

clip_image149

clip_image151

clip_image153

Example 9

Find the value of clip_image155

Solution

Once again, direct substitution yields, clip_image157. Therefore we need to rationalize the denominator of the given rational function before we evaluate.

clip_image159

clip_image161

clip_image163

clip_image165

clip_image167

clip_image169

clip_image171

clip_image173

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