# What is the limit of x(sqrt(x^(2)+1))  as x approaches infinity?

Jun 10, 2018

The limit $x \to \infty , f \left(x\right) \to \infty$

#### Explanation:

As $x$ approaches infinity the +1 will lose all significance so we can just consider:

$x \left(\sqrt{{x}^{2}}\right) = x \left(x\right) = {x}^{2}$

we know $x \to \infty , {x}^{2} \to \infty$ so we can conclude:

$f \left(x\right) = x \left(\sqrt{{x}^{2} + 1}\right)$

The limit $x \to \infty , f \left(x\right) \to \infty$

Jun 10, 2018

${\lim}_{x \to \infty} x \left(\sqrt{{\underbrace{{x}^{2}}}_{\text{humungous")+underbrace(1)\_("just 1}}}\right)$

$= {\lim}_{x \to \infty} x \cdot {\underbrace{\sqrt{{x}^{2}}}}_{\text{a positive number}}$

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