Help me with 2 Lim tasks

Nov 2017
1
0
Lithuania
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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skipjack

Forum Staff
Dec 2006
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2,470
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.
 
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v8archie

Math Team
Dec 2013
7,712
2,682
Colombia
First one: multiply by $$\frac{\sqrt{7+4x}+\sqrt{7-x}}{\sqrt{7+4x}+\sqrt{7-x}} \cdot \frac{\sqrt{3+x}+\sqrt{3+2x}}{\sqrt{3+x}+\sqrt{3+2x}}$$
Second one: write $2x-1=1+2(x-1)$ and use $$\lim_{t \to 0} (1+at)^{\frac1t}=e^a$$
 
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