How do you know or find if there is a hole in a rational function?

Question

luigi0210 · Answer

When there are like equations in the denominator and numerator, such as:
$$\frac{x-2}{(x-1)(x-2)}=0$$

The x-2 would cancel leaving a hole at x=2 and the new function would be:
$$\frac{1}{x-1}$$

tkhunny · Answer

I like to think of it this way:

1) If a value for x makes the denominator zero (0), it is a POTENTIAL vertical asymptote.
2) If that value for x also makes the numerator zero (0), it is a hole, and not an asymptote at all.

tkhunny · Answer

I also like to think of it this way, using Luigi's example:

These two functions are EXACTLY the same

$y = \dfrac{(x-2)}{(x-1)(x-2)}$

$y = \dfrac{1}{x-1}$

EXCEPT at x = 2 where the first one fails to exist.

goformit100 · Answer

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