MCQ
$\mathop {Limit}\limits_{x\,\, \to \,\,0} $ ${\left( {\cos \,2x} \right)^{3\,/\,{x^2}}}$ has the value equal to ______ .
- A$e^{-3}$
- B$e^{-4}$
- C$e^{-5}$
- ✓$e^{-6}$
But:
$\cos ^{\frac{3}{x^{2}}}(2 x)=e^{\frac{3}{x^{2}} \ln |\cos (2 x)|}$ (have a look at the properties of logarithms)
and:
$\lim _{x \rightarrow 0} e^{\frac{3}{x^{2}} \ln [\cos (2 x) \mid}=e^{-6}$
The exponent $\frac{3}{x^{2}} \ln [\cos (2 x)]$ tends to -6
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