Question

What is the `lim_(x to oo) (2^x + 3^x)/(1 + 3^x)`?

Answer 1

Given: `lim_(x to oo) ( 3^x+ 2^x)/(3^x+ 1)`

Divide numerator and denominator by the denominator's leading term:

`lim_(x to oo) ( 1+ (2/3)^x)/(1+ (1/3)^x)`

We know that the limit of any number less than 1 to the power of x goes to 0 as x goes to infinity:

`( 1+ (2/3)^oo)/(1+ (1/3)^oo) = (1+ 0)/(1+0) = 1`

Therefore, the original limit is 1:

`lim_(x to oo) ( 3^x+ 2^x)/(3^x+ 1) = 1`