evaluate the limit below in two steps by using algebra to simplify the difference quotient and then…

evaluate the limit below in two steps by using algebra to simplify the difference quotient and then evaluating the limit. $lim_{h\rightarrow0^{+}}left(\frac{sqrt{h^{2}+9h + 2}-sqrt{2}}{h}\right)=lim_{h\rightarrow0^{+}}left(square\right)=square$

evaluate the limit below in two steps by using algebra to simplify the difference quotient and then evaluating the limit. $lim_{h\rightarrow0^{+}}left(\frac{sqrt{h^{2}+9h + 2}-sqrt{2}}{h}\right)=lim_{h\rightarrow0^{+}}left(square\right)=square$

Answer

Explanation:

Step1: Rationalize the numerator

Multiply the fraction by $\frac{\sqrt{h^{2}+9h + 2}+\sqrt{2}}{\sqrt{h^{2}+9h + 2}+\sqrt{2}}$. [ \begin{align*} &\lim_{h\rightarrow0^{+}}\frac{\sqrt{h^{2}+9h + 2}-\sqrt{2}}{h}\times\frac{\sqrt{h^{2}+9h + 2}+\sqrt{2}}{\sqrt{h^{2}+9h + 2}+\sqrt{2}}\ =&\lim_{h\rightarrow0^{+}}\frac{(h^{2}+9h + 2)-2}{h(\sqrt{h^{2}+9h + 2}+\sqrt{2})}\ =&\lim_{h\rightarrow0^{+}}\frac{h^{2}+9h}{h(\sqrt{h^{2}+9h + 2}+\sqrt{2})} \end{align*} ]

Step2: Simplify the fraction and evaluate the limit

Cancel out the common - factor $h$ in the numerator and denominator. [ \begin{align*} &\lim_{h\rightarrow0^{+}}\frac{h^{2}+9h}{h(\sqrt{h^{2}+9h + 2}+\sqrt{2})}\ =&\lim_{h\rightarrow0^{+}}\frac{h(h + 9)}{h(\sqrt{h^{2}+9h + 2}+\sqrt{2})}\ =&\lim_{h\rightarrow0^{+}}\frac{h + 9}{\sqrt{h^{2}+9h + 2}+\sqrt{2}} \end{align*} ] Substitute $h = 0$ into the expression: $\frac{0 + 9}{\sqrt{0^{2}+9\times0+2}+\sqrt{2}}=\frac{9}{2\sqrt{2}}=\frac{9\sqrt{2}}{4}$.

Answer:

$\frac{9\sqrt{2}}{4}$