# How do you find lim (x+1)^(3/2)-x^(3/2) as x->oo?

Jan 17, 2018

$\infty$ or the graph of the function will get undoubtedly large.

#### Explanation:

So we have ${\lim}_{x \to \infty} {\left(x + 1\right)}^{\frac{3}{2}} - {x}^{\frac{3}{2}}$

We can see that this function is continuous when $x \ge 0$
Remember that when a function is continuous at $c$, then ${\lim}_{x \to c} f \left(x\right) = f \left(c\right)$
So let's substitute $\infty$ in the place of $x$.

${\left(\infty + 1\right)}^{\frac{3}{2}} - {\infty}^{\frac{3}{2}}$

Now how do we solve that?

Well, let's use logic here.
If there is this really,really large number, and we are raising it to a power greater than one, will get an answer even greater than what we started with. Also, this function will give a positive value for any $x$ values that are equal to or greater than one.

Therefore, our function will get undoubtedly large as $x$ approaches infinity.

So you can say that ${\lim}_{x \to \infty} {\left(x + 1\right)}^{\frac{3}{2}} - {x}^{\frac{3}{2}}$ is $\infty$ or that it gets undoubtedly large.

We can even look at the graph of our function.
graph{(x+1)^(3/2)-x^(3/2) [-10, 10, -5, 5]}
Even though the rate that this is increasing is decreasing, there is no limit of how much the $y$ value can be.