How do you find the limit of $[(1/(x+1))-1]/x$ as x approaches 0?

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Answer 1 · 2015-08-01T18:08:54

$lim_(x->0)(( 1/(x+1) -1)/x) = -1$

Evaluating this limit for $x->0$ would result in the $0/0$ indeterminate form. To avoid this, focus on the numerator of this fraction

$1/(x+1) - 1$

This expression can be rewritten like this

$1/(x+1) - (x+1)/(x+1) = [1 - (x+1)]/(x+1) = (color(red)(cancel(color(black)(1))) - x + color(red)(cancel(color(black)(1))))/(x+1)$

$1/(x+1) - 1 = (-x)/(x+1)$

The limit will now be

$lim_(x->0)( (-color(red)(cancel(color(black)(x))))/(x+1) * 1/color(red)(cancel(color(black)(x)))) = lim_(xrarr0)-1/(x+1)$

For $x->0$ , this will result in

$lim_(x->0)(-1/(0+1)) = color(green)(-1)$

How do you find the limit of [(1/(x+1))-1]/x[(1/(x+1))-1]/x as x approaches 0?