# How do you find lim (1+5/sqrtu)/(2+1/sqrtu) as u->0^+ using l'Hospital's Rule?

May 5, 2017

5

May 5, 2017

${\lim}_{u \rightarrow {0}^{+}} \frac{1 + \frac{5}{\sqrt{u}}}{2 + \frac{1}{\sqrt{u}}} = 5$

#### Explanation:

We do not need to apply L'Hôpital's as this is a trivial limit to evaluate:

${\lim}_{u \rightarrow {0}^{+}} \frac{1 + \frac{5}{\sqrt{u}}}{2 + \frac{1}{\sqrt{u}}} = {\lim}_{u \rightarrow {0}^{+}} \frac{1 + \frac{5}{\sqrt{u}}}{2 + \frac{1}{\sqrt{u}}} \cdot \frac{\sqrt{u}}{\sqrt{u}}$
$\text{ } = {\lim}_{u \rightarrow {0}^{+}} \frac{\sqrt{u} + 5}{2 \sqrt{u} + 1}$
$\text{ } = \frac{5}{1}$

Note that the above limit is only valid for $u \rightarrow {0}^{+}$ as:

$u \rightarrow {0}^{+} \implies u > 0 \implies \sqrt{u} \in \mathbb{R}$

Whereas:

$u \rightarrow {0}^{-} \implies u < 0 \implies \sqrt{u} \in \mathbb{C}$