# What is the limit of  sqrt[x^3-4x^2+1] + sqrt[x^2-x] +x as x goes to infinity?

Sep 17, 2015

${\lim}_{x \to \infty} \sqrt{{x}^{3} - 4 {x}^{2} + 1} + \sqrt{{x}^{2} - x} + x = \infty$

#### Explanation:

${\lim}_{x \to \infty} \sqrt{{x}^{3} - 4 {x}^{2} + 1} = \infty$

${\lim}_{x \to \infty} \sqrt{{x}^{2} - x} = \infty$

${\lim}_{x \to \infty} x = \infty$

Since the limit of each term is $\infty$, the limit of the sum is $\infty$

${\lim}_{x \to \infty} \sqrt{{x}^{3} - 4 {x}^{2} + 1} + \sqrt{{x}^{2} - x} + x = \infty$

If you want to give justification for the first 2 limits, use

$\sqrt{{x}^{3} - 4 {x}^{2} + 1} = \sqrt{{x}^{3}} \sqrt{1 - \frac{4}{x} + \frac{1}{x} ^ 3}$

and $\sqrt{{x}^{2} - x} = \sqrt{{x}^{2}} \sqrt{1 - \frac{1}{x}}$.

In each case, as $x \to \infty$ the first factor also $\to \infty$ and the second goes to $1$