## anonymous one year ago find lim as x approaches 4 of ((sqrt(3x+4)-sqrt(4x))/(x^2-4x)

1. dan815

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2. dan815

like this?

3. anonymous

yes

4. dan815

both the top and bottom are going to 0, so you can apply l'hopitals rule

5. dan815

differentiate the top and differentiate the bottom separetly

6. dan815

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7. dan815

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8. anonymous

duh I forgot that square roots are another form of power to the half

9. IrishBoy123

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10. IrishBoy123

.....and a question the numerator isn't always 0 $$\sqrt{16} - \sqrt{16} = \pm4 \ - \pm 4 \implies -8, 0 , 8$$

11. myininaya

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.

12. dan815

thats a good point irish

13. dan815

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14. IrishBoy123

@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

15. IrishBoy123

like a conjugate, i guess, but having the nous to know that it works for the denominator too.

16. myininaya

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

17. IrishBoy123

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

18. myininaya

algebraic tricks aren't always easy to see so knowing l'hospital is a good back up plan or a first plan whatever

19. IrishBoy123

pearls of wisdom

20. IrishBoy123

good night.