Question

$EXPLANATION.\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to 0} \dfrac{sin(2 + x) - sin(2 - x)}{x}⟹limx→0​xsin(2+x)−sin(2−x)​\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ As we knw that,\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ First we put the value of x = 0 in equation, and check their indeterminant form, we get.\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to 0} \dfrac{sin(2 + 0) - sin(2 - 0)}{0}⟹limx→0​0sin(2+0)−sin(2−0)​\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to 0} \dfrac{sin(2) - sin(2)}{0}⟹limx→0​0sin(2)−sin(2)​\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to 0} \dfrac{0}{0}⟹limx→0​00​\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ As we can see that,\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ It is the form of 0/0 indeterminant form,\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ We can simply apply L HOSPITAL'S RULE.\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to 0} \dfrac{sin(2 + x) - sin(2 - x)}{x}⟹limx→0​xsin(2+x)−sin(2−x)​\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ Differentiate w.r.t x, we get.\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ ⇒ sin(2 + x).\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ ⇒ d(sin(2 + x)/dx\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ ⇒ cos(2 + x).1.\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ ⇒ cos(2 + x).\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ Differentiate w.r.t x, we get.\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ ⇒ sin(2 - x).\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ ⇒ d(sin(2 - x))/dx.\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ ⇒ cos(2 - x)(-1).\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ ⇒ -cos(2 - x).\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ We can write equation as,\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to 0} \dfrac{\dfrac{d(sin(2 + x))}{dx}\ \ - \dfrac{d(sin(2 - x))}{dx} }{\dfrac{d(x)}{dx} }⟹limx→0​dxd(x)​dxd(sin(2+x))​  −dxd(sin(2−x))​​\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to 0} \dfrac{cos(2 + x) - (-cos(2 - x))}{1}⟹limx→0​1cos(2+x)−(−cos(2−x))​\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to 0} = {cos(2 + x) + cos(2 - x)}⟹limx→0​=cos(2+x)+cos(2−x)\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ Put the value of x = 0 in equation, we get.\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to 0} = cos(2 + 0) + cos(2 - 0)⟹limx→0​=cos(2+0)+cos(2−0)\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to 0} = cos(2) + cos(2)⟹limx→0​=cos(2)+cos(2)\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to 0} = 2cos(2)⟹limx→0​=2cos(2)\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \                                                                                                                  \ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ MORE INFORMATION.\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ NOTE:\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ If function takes any of the following form 0/0 , ∞/∞ then L'HOSPITAL'S RULE applied.\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ \sf \implies \lim_{x \to a} \dfrac{f(x)}{g(x)} = \lim_{x \to a} \dfrac{f'(x)}{g'(x)}⟹limx→a​g(x)f(x)​=limx→a​g′(x)f′(x)​\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ Note :\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ L'HOSPITAL'S RULE can be repeated required number of times in same questions.\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater$​

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Light Questions

As light from a star spreads out and weakens, do gaps form between the photons?

Can air make shadows?

Can humans ever directly see a photon?

Can light bend around corners?

Can momentum be hidden to human eyes like how kinetic energy can be hidden as heat?

Can one bit of light bounce off another bit of light?

I'm sorry i can't get ur question

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