MCQ
$\mathop {\lim }\limits_{h \to 0} \frac{{\sqrt {x + h} - \sqrt x }}{h} = $
- ✓$\frac{1}{{2\sqrt x }}$
- B$\frac{1}{{\sqrt x }}$
- C$2\sqrt x $
- D$\sqrt x $
Aliter : Apply $L-$ Hospital rule,
$\mathop {\lim }\limits_{h \to 0} \,\,\frac{{\sqrt {x + h} - \sqrt x }}{h} = \mathop {\lim }\limits_{h \to 0} \,\,\frac{1}{{2\sqrt {x + h} }} = \frac{1}{{2\sqrt x }}$.
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$S=\left\{X \in R ^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\} \text { and }$
$T=\left\{Y \in R ^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\}.$
Then which of the following statements is (are) $TRUE$?
$(A)$ There is a triangle whose area is $1$ and all of whose vertices are from $S$.
$(B)$ There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $L M$ is also in $T$.
$(C)$ There are infinitely many rectangles of perimeter $48$ , two of whose vertices are from $S$ and the other two vertices are from $I$.
$(D)$ There is a square of perimeter $48$ , two of whose vertices are from $S$ and the other two vertices are from $T$.