Question

Lim [(1/x-1)-(2/x^2-1)], x->1 Right answer is 1/2

Answer 1

As you've probably noticed by now, both fractions' denominators turn out to be 0 when x = 1, so direct substitution is no good right now. However, we can combine the fractions first:


`(1)/(x - 1) - (2)/(x^2 - 1) &= (x + 1)/((x - 1)(x + 1)) - (2)/((x - 1)(x + 1)) = (x - 1)/((x - 1)(x + 1))`

Notice now that we can safely cancel out the x - 1 term because, since this is a limit, we only care about what happens near 1 instead of at 1, and the fraction is defined near 1. Doing so gives the fraction
`(1)/(x + 1)`, and substituting in 1 for x gives the answer
`\bf (1)/(2)`.