Question

Explain why the function f(x)=5x-15/x-3 is not asymptotic to the line x=3.Sketch the graph of this function

Answer 1

Answer:

We define an asymptote as a tendency to a given value, such that the value is never reached. We usually have vertical asymptotes in situations where the denominator of a function becomes zero.

We will see that we do not have an asymptote at x = 3 because the denominator can be simplified and the function is actually a constant.

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Let's see how to get the answer.

Here we start with the function:

f(x) = \frac{5x - 15}{x - 3}f(x)=

x−3

5x−15

And we want to see why this function does not have an asymptote at x = 3 (where the denominator is equal to zero).

This can happen because the denominator can be simplified.

If we write the numerator as:

5*x - 15 = 5*(x - 3)

Then we get:

f(x) = \frac{5*x - 15}{x - 3} = \frac{5*(x - 3)}{(x - 3)} = 5f(x)=

x−3

5∗x−15

=

(x−3)

5∗(x−3)

=5

So the numerator also became 0 when the denominator was zero, that is why we didn't have an asymptote at x = 3.