How do you find the limit of $\frac{\sqrt{x + 4} - 2}{x}$ $(sqrt(x+4) -2) / x$ as x approaches 0?

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Answer 1 · 2016-08-13T17:46:44

$\frac{1}{4}$ $1/4$

We have limit of indeterminate form, ie $\frac{0}{0}$ $0/0$ so can use L'Hopital's rule:

$lim_(xrarr0) (sqrt(x+4) - 2)/x = lim_(xrarr0)(d/(dx)(sqrt(x+4)-2))/(d/(dx)(x))$

$=lim_(xrarr0)(1/(2sqrt(x+4)))/1 = 1/(2sqrt(0+4)) = 1/4$

How do you find the limit of (sqrt(x+4) -2) / x√x+4−2x(sqrt(x+4) -2) / x as x approaches 0?