Question

limit as x approaches 0 (x^2+3x)(cotx)

Answer 1

\[\lim_{x \rightarrow 0}(x^2+3x)(cotx)\]

Answer 2

\[\lim_{x\rightarrow0}[(x^2+3x)*cot(x)]=\lim_{x\rightarrow 0}\frac{cos(x)x^2+3xcos(x)}{sin(x)}\\\overset{\color{blue}{H}}=\lim_{x\rightarrow 0}\frac{[-\sin(x)x^2+2x\cos(x)]+[3\cos(x)-\sin(x)3x]}{\cos(x)}\\\text{now plug in 0 because we are well defined there}\\\frac{0+0+3-0}{1}=3\]

Answer 3

Alternatively:
\[\lim_{x\to0}\left(x^2+3x\right)~\cot x=\lim_{x\to0}\frac{x^2+3x}{\tan x}\]
For \(x\) near 0, you have \(\tan x\approx 0\), so
\[\lim_{x\to0}\frac{x^2+3x}{\tan x}=\lim_{x\to0}\frac{x^2+3x}{x}=\lim_{x\to0}~\left(x+3\right)\]