Question

How to solve this limit?

Answer 1

(2x^2/(1-x^2)) + 3^(1/x)  for the first half you can simply divide all terms by the highest power of x to get:

2/(1/x^2-1)  as x approaches infinity you have 2/-1=-2

For the second part the exponent approaches zero as x approaches infinity and anything raised to the zero power is equal to 1 so you have:

-2+1=-1

So the limit as x approaches infinity is -1.

Answer 2

lim [(2x²/(1-x²) +3¹/ˣ] = lim (2x²/(1-x²) +lim 3¹/ˣ
x→∞                            x→∞                x→∞    

  Let's find the 2nd limit  : lim 3¹/ˣ = 3¹/∞ = 3⁰ = 1
                                        x→∞    

Now the 1st limit: lim (2x²/(1-x²): Divide numerator and denominator by x²:
                             x→∞   

  (2x²/(1-x²) = 2/(1/x² -x²/x²) = 2/(1/x² - 1)

When x→∞  2/(1/x² - 1) → 2/(1/∞ -1) = 2/(-1) = -2

Then lim [(2x²/(1-x²) +3¹/ˣ] = -1
         x→∞