usando o bizu da fatoração.
Temos :
[tex]\displaystyle \lim_{\text x \to +\infty}\frac{\sqrt{\text x}+1}{\text x+3} \\\\\\ \lim_{\text x\to +\infty }\frac{(\text x)^{\frac{1}{2}}+1}{\text x+3}[/tex]
vamos dividir a expressão pelo x que tem o maior grau, no caso é o próprio x.
[tex]\displaystyle \lim_{\text x\to +\infty}\frac{\displaystyle \frac{\text x^{\frac{1}{2}}}{\text x}+\frac{1}{\text x}}{\displaystyle \frac{\text x}{\text x}+\frac{3}{\text x}} \\\\\\ \lim_{\text x\to +\infty } \frac{\text x^{\displaystyle (\frac{1}{2}-1)}+\displaystyle \frac{1}{\text x}}{\displaystyle 1+\frac{3}{\text x}} \\\\\\ \lim_{\text x\to +\infty } \frac{\displaystyle \frac{1}{\sqrt{\text x}}+\frac{1}{\text x}}{\displaystyle 1+\frac{3}{\text x}}\\ \\\\ \underline{\text{fazendo x} \to +\infty} : \\\\[/tex]
[tex]\displaystyle \frac{\displaystyle \frac{1}{\sqrt{\infty }}+\frac{1}{\infty}}{\displaystyle 1+\frac{3}{\infty }} \\\\\\ \frac{0+ 0}{1+0} = 0 \\\\ \underline{\text{portanto}} : \\\\\\ \huge\boxed{\lim_{\text x\to \infty } \frac{\sqrt{\text x}+1}{\text x+3} = 0\ }\checkmark[/tex]
