Square roots are always causing trouble for me - especially finding the limit of a function when it's in the indeterminate form.
I got this far, but I don't know what to do next or even if it's right. Trying to go any further gives me my original problem.
This is the initial problem
lim
This is where I've gotten
\lim_{x\to5}\frac{\sqrt{x^2+5}-\sqrt{30}}{(x-5)} * \frac{\sqrt{x^2+5}+\sqrt{30}}{\sqrt{x^2+5}+\sqrt{30}} = \frac{x^2+5-30}{(x-5)\left(\sqrt{x^2+5}+\sqrt{30}\right)}
Answer
HINT \ \ \ Factoring both of\ \ \sqrt{x^2+5}^{\:2} - \sqrt{30}^{\:2}\ =\ x^2 - 5^2\ by difference of squares yields
\frac{\sqrt{x^2+5}-\sqrt{30}}{x-5}\ =\ \frac{x+5}{\sqrt{x^2+5}+\sqrt{30}}
Alternatively, if you're familiar with derivatives, note that the limit is f\:'(5) for f(x) = \sqrt{x^2+5}\:.
Many limits can be simply calculated by recognizing them as instances of first derivatives and then calculating the derivatively rotely using known derivative rules. You can find a handful of examples in my prior posts starting here. Later in your course you will most likely encounter a generalization of this observation known as l'Hôpital's Rule.
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