Question

How do you find the limit of `(2x+1)^4/(3x^2+1)^2` as `x->oo`?

Answer 1

`lim_(xrarroo)(2x+1)^4/(3x^2+1)^2=16/9`

Explanation:

We don't have to expand these binomials. Factor out the greatest `x` degree from the numerator and denominator.

`lim_(xrarroo)(2x+1)^4/(3x^2+1)^2=lim_(xrarroo)[x(2+1/x)]^4/[x^2(3+1/x^2)]^2`

Distributing the exponents:

`=lim_(xrarroo)(x^4(2+1/x)^4)/(x^4(3+1/x^2)^2)=lim_(xrarroo)(2+1/x)^4/(3+1/x^2)^2`

As `xrarroo`, both `1/xrarr0` and `1/x^2rarr0`, since the denominator will be very large.

`=(2+0)^4/(3+0)^2=2^4/3^2=16/9`