Single Variable Calculus, Chapter 2, 2.3, Section 2.3, Problem 32

(a) Estimate the value of $\displaystyle \lim \limits_{x \to 0} \frac{\sqrt{3+x}-\sqrt{3}}{x}$ by using the graph of $f(x)$






Based on the graph, the limit of $f(x)$ seems to have a value of 0.29 as $x$ approaches 0.

(b) Guess the limit the limit by using a table of values of $f(x)$


$\begin{array}{|c|c|c|c|c|} \hline\\ x & 0.01 & 0.02 & 0.03 & 0.04 \\ \hline f(x) & 0.2884 & 0.2882 & 0.2879 & 0.2877\\ \hline \end{array}$

Based on the values from the table, the limit of the function seems to have a value of 0.29 as $x$ approaches to 0.

(c) Find the exact value of the limit using the limit laws.


$\begin{equation} \begin{aligned} & \lim\limits_{x \to 0} \frac{\sqrt{3 + x} - \sqrt{3}}{x} \cdot \frac{\sqrt{3 + x} + \sqrt{3}} {\sqrt{3 + x} + \sqrt{3}} = \lim \limits_{x \to 0} \frac{3 + x - 3}{ x( \sqrt{3 + x} + \sqrt{3} ) } && \text{ Simplify the equation by multiplying both numerator and denominator by } \sqrt{3 + x} + \sqrt{3} \\ &\lim \limits_{x \to 0} \frac{1}{\sqrt{3 + x} + \sqrt{3}} = \frac{\lim \limits_{x \to 0} 1 }{\lim \limits_{x \to 0} \sqrt{3 + x} + \lim \limits_{x \to 0} \sqrt{3}} && \text{ Quotient and root law.} \\ & \lim \limits_{x \to 0}\frac{1}{\sqrt{3 + x} + \sqrt{3}} = \frac{1}{\sqrt{\lim \limits_{x \to 0}(3 + x)} + \sqrt{3}} && \text{ Constant and root law.}\\ & \lim \limits_{x \to 0} \frac{1}{\sqrt{3 + x} + \sqrt{3}} = \frac{1 }{\sqrt{3 + 0} + \sqrt{3}} && \text{ Sum, constant and special limit law.}\\ \end{aligned} \end{equation}\\ \boxed{\displaystyle \lim \limits_{x \to 0} \frac{1}{\sqrt{3 + x} + \sqrt{3}} = \frac{1}{2 \sqrt{3}}}$