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Question

Evaluate the following limits.
$$\displaystyle\lim_{x\rightarrow a}\dfrac{\cos x-\cos a}{\sqrt{x}-\sqrt{a}}$$.


Solution

$$\begin{matrix} { \lim   }_{ x\to a }\frac { { \cos  x-\cos  a } }{ { \sqrt { x } -\sqrt { a }  } }  \\ It\, \, is\, \, \frac { 0 }{ 0 } \, \, form\,  \\ use\, \, { L_{ 1 } }\, \, Hospital\, \, rule. \\ { \lim   }_{ x\to a }\frac { { -\sin  x-0 } }{ { \frac { 1 }{ { 2\sqrt { x }  } } -0 } } =-2\sqrt { x } \sin  x \\ { \lim   }_{ x\to a }=-2\sqrt { a } \sin  a\, \, Ans. \\  \end{matrix}$$

Mathematics

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