Use a CAS to plot the function near the point x0 being approached. From your plot guess the value of the limit.
-$$\lim _ { x \rightarrow 0 } \frac { \sqrt { 10 + 10 x } - \sqrt { 10 } } { x }$$
A) $\frac { \sqrt { 10 } } { 2 }$
B) 0
C) $\sqrt { 10 }$
D) $\frac { 1 } { 2 }$

Question

Quizplus · Accepted Answer

Using a CAS such as Desmos or Wolfram Alpha, we can plot the function near $x=0$:

[asy]
size(200);
import TrigMacros;
rr_cartesian_axes(-1, 1, -1, 3,complexplane=false,usegrid=false);
pen thin=linewidth(0.7); 
real f(real x) { return (sqrt(10+10x)-sqrt(10))/x; }
draw(reflect((0,0),(1,1))*graph(f,-1,1, n=400,join=operator ..),red+thin);
[/asy]

From the graph, we see that the function appears to approach a constant value as $x$ gets closer to 0. Furthermore, the limit appears to be positive (since the function appears to be approaching a positive value). The only answer choice that satisfies these criteria is $\boxed{	extbf{(A) } \frac{\sqrt{10}}{2}}$. We can also algebraically verify that this is the correct answer by using rationalizing the numerator: \begin{align*}
\frac{\sqrt{10+10x}-\sqrt{10}}{x} &= \frac{\sqrt{10+10x}-\sqrt{10}}{x}\cdot\frac{\sqrt{10+10x}+\sqrt{10}}{\sqrt{10+10x}+\sqrt{10}} \
&= \frac{10x}{x(\sqrt{10+10x}+\sqrt{10})} \
&= \frac{10}{\sqrt{10+10x}+\sqrt{10}}
\end{align*} As $x	o 0$, we have $\sqrt{10+10x}	o \sqrt{10}$ and so $\frac{10}{\sqrt{10+10x}+\sqrt{10}} 	o \frac{10}{2\sqrt{10}}=\frac{\sqrt{10}}{2}$.

Use a CAS to Plot the Function near the Point lim⁡x→010+10x−10x\lim _ { x \rightarrow 0 } \frac { \sqrt { 10 + 10 x } - \sqrt { 10 } } { x }x→0lim​x10+10x​−10​​

Use a CAS to Plot the Function near the Point $\lim _ { x \rightarrow 0 } \frac { \sqrt { 10 + 10 x } - \sqrt { 10 } } { x }$