lim _{x rightarrow 0} frac{15^{x}-5^{x}-3^{x} | vedclass

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Question

(A) આપેલ લક્ષ: $\lim _{x \rightarrow 0} \frac{15^{x}-5^{x}-3^{x}+1}{1-\cos 2 x}$ અંશનું અવયવીકરણ કરતા: $15^{x}-5^{x}-3^{x}+1 = 5^{x}(3^{x}-1) - 1(3^{x

Vedclass · Accepted Answer

$\frac{(\log 3)(\log 5)}{2}$

Vedclass · Answer

(A) આપેલ લક્ષ: $\lim _{x \rightarrow 0} \frac{15^{x}-5^{x}-3^{x}+1}{1-\cos 2 x}$ અંશનું અવયવીકરણ કરતા: $15^{x}-5^{x}-3^{x}+1 = 5^{x}(3^{x}-1) - 1(3^{x}-1) = (3^{x}-1)(5^{x}-1)$ નિત્યસમ $1-\cos 2x = 2\sin^{2}x$ નો ઉપયોગ કરતા: $\lim _{x \rightarrow 0} \frac{(3^{x}-1)(5^{x}-1)}{2\sin^{2}x}$ અંશ અને છેદને $x^{2}$ વડે ભાગતા: $\lim _{x \rightarrow 0} \frac{(\frac{3^{x}-1}{x})(\frac{5^{x}-1}{x})}{2(\frac{\sin x}{x})^{2}}$ પ્રમાણિત લક્ષ $\lim _{x \rightarrow 0} \frac{a^{x}-1}{x} = \log a$ અને $\lim _{x \rightarrow 0} \frac{\sin x}{x} = 1$ નો ઉપયોગ કરતા: $= \frac{(\log 3)(\log 5)}{2(1)^{2}} = \frac{(\log 3)(\log 5)}{2}$

$\lim _{x \rightarrow 0} \frac{15^{x}-5^{x}-3^{x}+1}{1-\cos 2 x}$ ની કિંમત શોધો.

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