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November 18th, 2017, 08:06 AM   #1
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Question Help me with 2 Lim tasks

Hello there. Could someone help me, and explain how these 2 tasks should be done. I added them in attachment. I know how to do simple lim tasks, but these ones are more complicated and I'm really not best at math.

Thank You.
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November 18th, 2017, 08:25 AM   #2
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The first limit is easily found by use of de l'Hôpital's rule.

For a proof that doesn't rely on calculus, the method suggested by v8archie below is best.
Thanks from topsquark and v8archie

Last edited by skipjack; November 18th, 2017 at 09:30 AM.
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November 18th, 2017, 08:32 AM   #3
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First one: multiply by $$\frac{\sqrt{7+4x}+\sqrt{7-x}}{\sqrt{7+4x}+\sqrt{7-x}} \cdot \frac{\sqrt{3+x}+\sqrt{3+2x}}{\sqrt{3+x}+\sqrt{3+2 x}}$$
Second one: write $2x-1=1+2(x-1)$ and use $$\lim_{t \to 0} (1+at)^{\frac1t}=e^a$$
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