Limits detailed solution

Question

The value of  limx→∞ (3^x + 3^2x)^(1/x)

Answer 1

Hi Richa.

 

I'm going to walk through a multi-step algebra nightmare involving exponentials. Hope you can follow along okay.

 

(3^x + 3^2x)^1/x

 

= (3^x(1 + 3^x))^1/x

 

= ((3^x)^1/x)(1 + 3^x)^1/x

 

= 3*(1 + 3^x)^1/x

 

= 3*(3^x(1/3^x + 1))^1/x

 

= 3*((3^x)^1/x)(1/3^x + 1)^1/x

 

= 3*3*(1/3^x + 1)^1/x

 

= 9*(1/3^x + 1)^1/x

 

As x --> ∞, the 1/3^x term is going to vanish to zero. The exponent 1/x will also vanish to 0, leaving you with just

 

9*(1)⁰

 

which is just 9*1

 

which is 9.

 

So

 

lim as x -->∞ (3^x + 3^2x)^1/x = 9

 

Let me know if any of the steps don't make sense. Hope this helps.