Question

${lim}_{x\to 0}\frac{x^2+1-cosx}{xsinx}$

Answer 1

${lim}_{x\to 0}\frac{x^2+1-cosx}{xsinx}$

${lim}_{x\to 0}\frac{x^2+(1-cosx)}{x^2\frac{sinx}{x}}$

${lim}_{x\to 0}\frac{x^2+2si{n}^2\frac{x}{2}}{x^2}[{lim}_{x\to 0}\frac{sinx}{x}=1\because ]$

$={lim}_{x\to 0}(1+\frac{2si{n}^2\frac{x}{2}}{x^2})$

$={lim}_{x\to 0}[1+2{\left(\frac{sin\frac{x}{2}}{\frac{x}{2}}\right)}^2\times \frac{1}{4}]$

$=1+\frac{1}{2}{lim}_{\frac{x}{2}\to 0}{\left(\frac{sin\frac{x}{2}}{\frac{x}{2}}\right)}^2$

$=1+\frac{1}{2}$

$=\frac{3}{2}$