find lim as x approaches 4 of ((sqrt(3x+4)-sqrt(4x))/(x^2-4x)

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find lim as x approaches 4 of ((sqrt(3x+4)-sqrt(4x))/(x^2-4x)

Mathematics
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both the top and bottom are going to 0, so you can apply l'hopitals rule
differentiate the top and differentiate the bottom separetly
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duh I forgot that square roots are another form of power to the half
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.....and a question the numerator isn't always 0 \(\sqrt{16} - \sqrt{16} = \pm4 \ - \pm 4 \implies -8, 0 , 8\)
i think this square root is the principal square root you could also do this without l'hospital you can rationalize the numerator and then which eventually leads to canceling the (x-4) factor out on top and bottom and then you will be able to do direct substitution.
thats a good point irish
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@myininaya thank you!! \(\large \frac{\sqrt{3x+4}-\sqrt{4x}}{x^2 - 4x} .\frac{\sqrt{3x+4}+\sqrt{4x}}{\sqrt{3x+4}+\sqrt{4x}}\) \(= \frac{3x+4-4x}{x(x-4)(\sqrt{3x+4}+\sqrt{4x})}\) \(= \frac{-(x-4)}{x(x-4)(\sqrt{3x+4}+\sqrt{4x})}\) \(= -\frac{1}{x(\sqrt{3x+4}+\sqrt{4x})}\) \(= -\frac{1}{4(\sqrt{16}+\sqrt{16})}\) \(= -\frac{1}{32}\) not familiar with this trick
like a conjugate, i guess, but having the nous to know that it works for the denominator too.
here is a fun one and sorta similar one: \[\lim_{x \rightarrow 2}\frac{\sqrt{6-x}-2}{\sqrt{3-x}-1} \\ \lim_{x \rightarrow 2} \frac{\sqrt{6-x}-2}{\sqrt{3-x}-1} \cdot \frac{\sqrt{3-x}+1}{\sqrt{3-x}+1} \cdot \frac{\sqrt{6-x}+2}{\sqrt{6-x}+2} \\ \lim_{x \rightarrow 2}\frac{ 6-x-4}{3-x-1} \frac{\sqrt{ 3-x}+1}{\sqrt{6-x}+2} \\ \lim_{x \rightarrow 2} \frac{-x+2}{-x+2} \frac{\sqrt{3-x}+1}{\sqrt{6-x}+2} \\ \lim_{x \rightarrow 2} \frac{\sqrt{3-x}+1}{\sqrt{6-x}+2} \\ = \frac{\sqrt{3-2}+1}{\sqrt{6-2}+2}=\frac{2}{2+2}=\frac{1}{1+1}=\frac{1}{2}\] I always thought the two conjugate thing was really cute for some reason
waoh! that's really cool way back then, when i first learned this stuff, you used l'Hopital when you had no other way out. it was a footnote. so thanks :p
algebraic tricks aren't always easy to see so knowing l'hospital is a good back up plan or a first plan whatever
pearls of wisdom
good night.

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