Find Discontinuity And Classifying Them

The problem I am looking at is $\displaystyle f(x) = \frac{x - 3}{x^2 - 9}$

I found that f was not defined at $\displaystyle x = \pm 3$

I have -3 as a non-removable discontinuity, and 3 as a removable discontinuity, approaching positive infinity from either side. Yet, when I graph the function, it is unequivocally approaching a finite value. Why did limiting process give me positive infinity?

Re: Find Discontinuity And Classifying Them

Quote:

Originally Posted by

**Bashyboy** The problem I am looking at is $\displaystyle f(x) = \frac{x - 3}{x^2 - 9}$

I found that f was not defined at $\displaystyle x = \pm 3$

I have -3 as a non-removable discontinuity, and 3 as a removable discontinuity, approaching positive infinity from either side. Yet, when I graph the function, it is unequivocally approaching a finite value. Why did limiting process give me positive infinity?

Notice that if $\displaystyle \displaystyle \begin{align*} x \neq 3 \end{align*}$, we have

$\displaystyle \displaystyle \begin{align*} \frac{x - 3}{x^2 - 9} &\equiv \frac{x - 3}{(x - 3)(x + 3)} \\ &\equiv \frac{1}{x + 3} \end{align*}$

The left hand limit as $\displaystyle \displaystyle \begin{align*} x \to -3 \end{align*}$ is $\displaystyle \displaystyle \begin{align*} -\infty \end{align*}$ while the right hand limit as $\displaystyle \displaystyle \begin{align*} x \to -3 \end{align*}$ is $\displaystyle \displaystyle \begin{align*} +\infty \end{align*}$. Since the left and right hand limits are not equal, this is a jump discontinuity.

Re: Find Discontinuity And Classifying Them

Oh, so you can cancel out factors; I thought I remembered my teacher saying we couldn't, or maybe that was just in the original function.

Re: Find Discontinuity And Classifying Them

Quote:

Originally Posted by

**Bashyboy** Oh, so you can cancel out factors; I thought I remembered my teacher saying we couldn't, or maybe that was just in the original function.

You can cancel out factors as long as you realise that there is an extra discontinuity in your original function.