Question

$$lim_{x\rightarrow0}\frac{\sqrt{1+x\sin\ x-1}}{x^{2}}$$

Answer

l<0

Solution


Simplify  \(1+x\sin{x}-1\)  to  \(x\sin{x}\).
\[l\imath mx->0\times \frac{\sqrt{x\sin{x}}}{{x}^{2}}\]
Simplify  \(0\times \frac{\sqrt{x\sin{x}}}{{x}^{2}}\)  to  \(0\).
\[l\imath mx->0\]
Regroup terms.
\[-+l\imath mx>0\]
Divide both sides by \(\imath \).
\[-+lmx>0\]
Divide both sides by \(m\).
\[-+lx>0\]
Divide both sides by \(x\).
\[-+l>0\]
Multiply both sides by \(-1\).
\[l<0\]