Question

If ${lim}_{x\to 0}kxcosecx={lim}_{x\to 0}xcoseckx$,find k.

Answer 1

${lim}_{x\to 0}kxcosecx={lim}_{x\to 0}xcoseckx$

LHS ${lim}_{x\to 0}kxcosecx=k{lim}_{x\to 0}xcosecx$

$=k{lim}_{x\to 0}\frac{x}{sinx}=k{lim}_{x\to 0}\frac{1}{\frac{sinx}{x}}$

$=k\times 1[\because {lim}_{\theta \to 0}\frac{sin\theta }{\theta }=1]$

=k

Also,RHS=${lim}_{x\to 0}xcoseckx$

$={lim}_{x\to 0}\frac{x}{sinkx}={lim}_{x\to 0}\frac{x}{\frac{sinkx}{kx}\times kx}$

$=\frac{1}{k}[\because {lim}_{\theta \to 0}\frac{sin\theta }{\theta }=1]$

As,LHS=RHS

$\Rightarrow k=\frac{1}{k}$

$k^2=1$

$k=\pm 1$