Answer:
A. [tex]\lim_{x \to 0} \frac{\sqrt{x+2}-\sqrt{2}}{x}=0.3535[/tex]
Step-by-step explanation:
We are given the limit expression, [tex]\lim_{x \to 0} \frac{\sqrt{x+2}-\sqrt{2}}{x}[/tex]
As, we see that,
When [tex]x\rightarrow 0[/tex], the function is of the form [tex]\frac{0}{0}[/tex].
So, we will use L'Hospital's Rule to proceed further i.e. Differentiate the numerator and denominator with respect to x.
That is,
[tex]\lim_{x \to 0} \frac{\sqrt{x+2}-\sqrt{2}}{x}[/tex]
implies [tex]\lim_{x \to 0} \frac{\frac{1}{2\sqrt{x+2}}}{1}[/tex]
i.e. [tex]\lim_{x \to 0} \frac{1}{2\sqrt{x+2}}}=\frac{1}{2\sqrt{2}}=0.3535[/tex]
Thus, [tex]\lim_{x \to 0} \frac{\sqrt{x+2}-\sqrt{2}}{x}=0.3535[/tex]
Hence, option A is correct.
