Find all discontinuities of f(x). For each discontinuity that is removable, define a new function that removes the discontinuity. f(x) = x - 1/x^2 - 1

Question

Find all discontinuities of f(x).  For each discontinuity that is removable, define a new function that removes the discontinuity. f(x) = x - 1/x^2 - 1

tkhunny · Answer

First, please remember your Order of Operations.  You have NOT written $\dfrac{x-1}{x^{2}-1}$.  Give it another go and use more parentheses.

Denominator = 0 -- Discontinuity.

Is it an Asymptote or NonRemovable Discoutinuity?

Numerator = 0 AT THE SAME PLACE, this it's Removable and NOT an Asymptote.

anonymous · Answer

non removable

tkhunny · Answer

??  There are two.

1) Which one are you talking about.
2) What's the other one?

anonymous · Answer

removable discontinuity or infinite

anonymous · Answer

so it will be replacing 0 will give you discontity right? and not any other number?

tkhunny · Answer

Please make a better effort to use complete sentences and to be substantially more clear.  I'll do a quick example.  Sentences, paragraphs, examples, order.

Working with the Denominator:

$x^{2} - 1 = (x+1)(x-1)$

This denominator takes on the value zero at x = 1 and x = -1.  These values are NOT in the Domain and are discontinuities.  We do not yet know what kind of discontinuity.

Working with the Numerator

x - 1 = 0 when x = 1

This is enough information.

x = 1 makes both Numerator and Denominator zero.  This is, therefore, a removable discontinuity.  Our original expression is equivalent to 1/(x+1) everywhere EXCEPT x = 1.

x = -1 makes only the denominator zero.  This is, therefore, an asymptote or infinite discontinuity.

Now, we are done.