Question

lim x-->13 1/x-13

How do you find the limit, if it exists?

Is direct substitution the only way to do this?

Answer 1

think of it this way:
what happens to 1/x-13 when x aproches 13 from the right or left (it doesn't matter in tis case), but never really equals to 13 . The denominator will get smaller and smaller. So the expression will tend to infinity.

Answer 2

So on a multiple choice question where infinity isn't an option but DNE is, the answer would be does not exist, right?

Answer 3

not quite ... limit of infinity is different than DNE

a limit only DNE if right and left side limits dont agree

to test this, plug in number on either side of 13 like 12.9 and 13.1

x=12.9 --> 1/-.1 = -10 , this means its approaching -infinty

x=13.1 --> 1/.1 = 10 , approaching pos infinity

the limits dont agree, thus overall limit DNE