Question

Evaluate
limit
x tends to 2 f(x)-f(2)/x-2 where f(x)=x2-4x​

Answer 1

☆Answer ☆

In order to evaluate a limit we are not interested in the value of the function at the limit, just the behaviour of the function around the limit:

We have:

f(x)={x2x≠22x=2

If we graphed the function would be the parabola y=x2 with a hole at x=2.

If we just examine the behaviour close to x=2, e.g x=2±0.001 we get:

f(2−0.001)=f(1.999)=3.996001

f(2+0.001)=f(2.001)=4.004001

f(2)=2

Which would certainly "suggest" that limx→2f(x) is 4 rather than 2. We can show thi is the case analytically as follows by studying the limit either side of x=2:

The Left Handed limit:

limx→2−f(x)=limx→2−x2

=22

=4

And The Right Handed limit:

limx→2+f(x)=limx→2+x2

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