<s>[INST] easy way to calculate the limit $\lim_{x \to 0 } \frac{\cos{x}- (\cos{x})^{\cos{x}}}{1-\cos{x}+\log{\cos{x}}}$ I have been trying to use L'Hôpital over this, but its getting too long, is there a short and elegant solution for this?
The Limit approaches 2 according to wolfram.
 [/INST] How to solve
$$
\begin{align*}
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\cos(x) - \cos(x)^{\cos(x)}}{1 - \cos(x) + \log(\cos(x))}\\
\lim_{{x} \to {0}} f(x) &= \frac{\cos(0) - \cos(0)^{\cos(0)}}{1 - \cos(0) + \log(\cos(0))}\\
\lim_{{x} \to {0}} f(x) &= \frac{1 - 1^{1}}{1 - 1 + \log(1)}\\
\lim_{{x} \to {0}} f(x) &= \frac{0}{0} \quad\mid\quad \text{we have to use L'hospital rule}\\
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&\text{if we use the rule of L'hospital one time, we'll get the same situation aigan and aigan aka let's take the rule of L'hospital three
times}\\
\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\cos(x) - \cos(x)^{\cos(x)}}{1 - \cos(x) + \log(\cos(x))}\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\frac{\mathrm{d^{3}}}{\mathrm{d}^{3}x} ~ (\cos(x) - \cos(x)^{\cos(x)})}{\frac{\mathrm{d^{3}}}{\mathrm{d}^{3}x} ~ (1 - \cos(x) + \log(\cos(x)))}\\
\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\frac{\mathrm{d}}{\mathrm{d}x} ~ (\cos(x) - \cos(x)^{\cos(x)})}{\frac{\mathrm{d}}{\mathrm{d}x} ~ (1 - \cos(x) + \log(\cos(x)))}\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\cos(x) \cdot (\sin(x) + \log(\cos(x)) \cdot \sin(x))}{\cos(x) - \tan(x)}\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\sin(x) - \cos(x)^{\cos(x)} \cdot -(\sin(x) + \log(\cos(x)) \cdot \sin(x))}{\cos(x) - \frac{\sin(x)}{\cos(x)}}\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\cos(x) \cdot (-1 + \cos(x)^{\cos(x)} + \cos(x)^{\cos(x)} \cdot \log(\cos(x)))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \cos(x) \cdot \lim_{{x} \to {0}} \frac{-1 + \cos(x)^{\cos(x)} + \cos(x)^{\cos(x)} \cdot \log(\cos(x))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= \cos(0) \cdot \lim_{{x} \to {0}} \frac{-1 + \cos(x)^{\cos(x)} + \cos(x)^{\cos(x)} \cdot \log(\cos(x))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= 1 \cdot \lim_{{x} \to {0}} \frac{-1 + \cos(x)^{\cos(x)} + \cos(x)^{\cos(x)} \cdot \log(\cos(x))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{-1 + \cos(x)^{\cos(x)} + \cos(x)^{\cos(x)} \cdot \log(\cos(x))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{-1 + \cos(x)^{\cos(x)}}{\cos(x) - 1} + \lim_{{x} \to {0}} \frac{\cos(x)^{\cos(x)} \cdot \log(\cos(x))}{\cos(x) - 1} \quad\mid\quad \text{since both the numerator and denominator appraoch to 0 } \frac{-1 + \cos(x)^{\cos(x)}}{\cos(x) - 1} \text{ is our canditate}\\
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\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\frac{\mathrm{d}}{\mathrm{d}x} ~ (-1 + \cos(x)^{\cos(x)})}{\frac{\mathrm{d}}{\mathrm{d}x} ~ (\cos(x) - 1)}\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \cos(x)^{\cos(x)} \cdot (\log(\cos(x)) + 1) + \lim_{{x} \to {0}} \frac{\cos(x)^{\cos(x)} \cdot \log(\cos(x))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= \cos(0)^{\cos(0)} \cdot (\log(\cos(0)) + 1) + \lim_{{x} \to {0}} \frac{\cos(x)^{\cos(x)} \cdot \log(\cos(x))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= 1^1 \cdot (\log(1) + 1) + \lim_{{x} \to {0}} \frac{\cos(x)^{\cos(x)} \cdot \log(\cos(x))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= 1 \cdot (0 + 1) + \lim_{{x} \to {0}} \frac{\cos(x)^{\cos(x)} \cdot \log(\cos(x))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= 1 + \lim_{{x} \to {0}} \frac{\cos(x)^{\cos(x)} \cdot \log(\cos(x))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= 1 + \lim_{{x} \to {0}} \cos(x)^{\cos(x)} \cdot \lim_{{x} \to {0}} \frac{\log(\cos(x))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= 1 + 1^{\cos(x)} \cdot 1 \frac{\log(\cos(x))}{\cos(x) - 1}\\
\lim_{{x} \to {0}} f(x) &= 1 + 1^{\cos(x)} \frac{\log(\cos(x))}{\cos(x) - 1}\\
\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\frac{\mathrm{d}}{\mathrm{d}x} ~ \log(\cos(x)))}{\frac{\mathrm{d}}{\mathrm{d}x} ~ (\cos(x) - 1)} + 1\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{-\tan(x)}{-\sin(x)} + 1\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\tan(x)}{\sin(x)} + 1\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{\frac{\sin(x)}{\cos(x)}}{\sin(x)} + 1\\
\lim_{{x} \to {0}} f(x) &= \lim_{{x} \to {0}} \frac{1}{\cos(x)} + 1\\
\lim_{{x} \to {0}} f(x) &= \frac{1}{\cos(0)} + 1\\
\lim_{{x} \to {0}} f(x) &= \frac{1}{1} + 1\\
\lim_{{x} \to {0}} f(x) &= 1 + 1\\
\lim_{{x} \to {0}} f(x) &= 2\\
\end{align*}
$$
Are there still questions?^^
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