MCQ
The value of $\mathop {\lim }\limits_{x \to 0} \,\frac{{{e^x} - x - 1}}{{{x^2}}}$ is
- ✓$0.5$
- B$0$
- C$1$
- D$-1$
This is an indeterminate type so use I'Hopital's Rule. That is, take the derivative of the top and the bottom and then find the limit of its quotient.
$\lim _{x \rightarrow 0} \frac{e^{x}-1}{2 x}=\frac{e^{0}-1}{0}=\frac{1-1}{0}=\frac{0}{0}$ This is still an indeterminate form so let's
use I'Hopital's Rule again.
$\lim _{x \rightarrow 0} \frac{e^{x}}{2}=\frac{e^{0}}{2}=\frac{1}{2}$
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$x\,\, + \,\,y\,\, = \,\,\frac{{2\pi }}{3},\,{\rm{cos}}\,{\rm{x + }}\,{\rm{ cos}}\,{\rm{y}}\,{\rm{ = }}\,\frac{3}{2},$ where $x$ and $y$ are real in