# What is the limit of (2x-3)^20(3x+2)^30 / (2x+1)^50 as x goes to infinity?

Sep 15, 2015

${\left(\frac{3}{2}\right)}^{30}$

#### Explanation:

${\left(2 x - 3\right)}^{20} = {2}^{20} {x}^{20} + \text{terms of degree lass than 20}$

${\left(3 x + 2\right)}^{30} = {3}^{30} {x}^{30} + \text{terms of degree lass than 30}$

So the numerator, when we expand, will be

${2}^{20} {3}^{30} {x}^{50} + \text{terms of degree less than 50}$

and

${\left(2 x + 1\right)}^{50} = {2}^{50} {x}^{50} + \text{terms of degree less than 50}$

The ratio is:

$\left({2}^{20} {3}^{30} {x}^{50} + \left(\text{terms of degree less than 50"))/(2^50 x^50 + ("terms of degree less than 50}\right)\right)$

Now, if we multiply by $\frac{\frac{1}{x} ^ 50}{\frac{1}{x} ^ 50}$, we get:

(2^20 3^30 + ("terms of degree less than 50")/x^50)/(2^50 + "terms of degree less than 50"/x^50)

As $x \rightarrow \infty$, this goes to $\frac{{2}^{20} {3}^{30}}{2} ^ 50 = {3}^{30} / {2}^{30}$