Question

evaluate the limit. estimate the value of lim x is approaching 0 (x)/√(1+3x)-1. b. make a table of values of f(x) for x close to 0 and guess the value of the limit.

Answer 1

\[\lim_{x\rightarrow 0}\frac{x}{\sqrt{1+3x}-1}=\lim_{x\rightarrow 0}\frac{x}{\sqrt{1+3x}-1}\cdot\frac{\sqrt{1+3x}+1}{\sqrt{1+3x}+1}=\]\[=\lim_{x\rightarrow 0}\frac{x(\sqrt{1+3x}+1)}{1+3x-1}=\frac{1}{3}\lim_{x\rightarrow 0}(\sqrt{1+3x}+1)=\frac{2}{3}\]