November 18th, 2017, 07:06 AM  #1 
Newbie Joined: Nov 2017 From: Lithuania Posts: 1 Thanks: 0  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. 
November 18th, 2017, 07:25 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,183 Thanks: 1647 
The first limit is easily found by use of de l'HÃ´pital's rule. For a proof that doesn't rely on calculus, the method suggested by v8archie below is best. Last edited by skipjack; November 18th, 2017 at 08:30 AM. 
November 18th, 2017, 07:32 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,330 Thanks: 2457 Math Focus: Mainly analysis and algebra 
First one: multiply by $$\frac{\sqrt{7+4x}+\sqrt{7x}}{\sqrt{7+4x}+\sqrt{7x}} \cdot \frac{\sqrt{3+x}+\sqrt{3+2x}}{\sqrt{3+x}+\sqrt{3+2 x}}$$ Second one: write $2x1=1+2(x1)$ and use $$\lim_{t \to 0} (1+at)^{\frac1t}=e^a$$ 

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