Question

what is th limit of (sqrt(25 + x) - 5)/x as x approaches 0

Answer 1

Hey Kmar,
\(\Large\bf \color{#CC0033}{\text{Welcome to OpenStudy! :)}}\)

Answer 2

\[\Large\rm \lim_{x\to0}\frac{\sqrt{25+x}-5}{x}\]

Answer 3

Multiply the numerator and denominator by the `conjugate of the numerator`,\[\Large\rm \lim_{x\to0}\frac{\sqrt{25+x}-5}{x}\color{royalblue}{\left(\frac{\sqrt{25+x}+5}{\sqrt{25+x}+5}\right)}\]

Answer 4

In the numerator, we are multiplying conjugates,\[\Large\rm (a-b)(a+b)=a^2-b^2\]So your square root should disappear up there, and you should get a bunch of nice cancellations and stuff.

\[\Large\rm \lim_{x\to0}\frac{(\sqrt{25+x}-5)(\sqrt{25+x}+5)}{x\left(\sqrt{25+x}+5\right)}\]Leave the bottom alone, don't distribute the x.

Answer 5

After you cancel some stuff out, you'll get to the point where you can directly plug in x=0 without getting an indeterminate form.