Evaluate without L'Hospital's rule or Series Expansion:
$\displaystyle \lim_{x\rightarrow 4} \frac{(\cos \alpha)^x - (\sin \alpha)^x - \cos 2\alpha}{x - 4}$
$\displaystyle cos^4 (\alpha) - sin^4 (\alpha) =(cos^2(\alpha) + sin^2(\alpha))(cos^2(\alpha)-sin^2(\alpha))=(cos^2(\alpha)-sin^2(\alpha))$
As $\displaystyle cos(2\alpha)=cos^2(\alpha)-sin^2(\alpha)$ the numerator is zero, and the denominator tends to zero... In this case the limit will be zero. But as x is not four, but tends to four, I'm not sure if the above trig. expressions can be used.
Hello,
That's correct til here.
0/0 is an undetermined limit !!!!!!As $\displaystyle cos(2\alpha)=cos^2(\alpha)-sin^2(\alpha)$ the numerator is zero, and the denominator tends to zero... In this case the limit will be zero. But as x is not four, but tends to four, I'm not sure if the above trig. expressions can be used.
Anyway, then you can notice that $\displaystyle \cos^4\alpha-\sin^4\alpha=\cos^2\alpha-\sin^2\alpha=\cos(2\alpha)$
So if you let $\displaystyle f(x)=\cos^x\alpha-\sin^x\alpha$, your limit is actually :
$\displaystyle \lim_{x\to 4} \frac{f(x)-f(4)}{x-4}$, which is (the difference quotient of the function f) equal to $\displaystyle f'(4)$
$\displaystyle \lim_{x\rightarrow 4} \frac{(\cos \alpha)^x - (\sin \alpha)^x - \cos 2\alpha}{x - 4}$
$\displaystyle =\lim_{x\rightarrow 4} \frac{(\cos \alpha)^x - (\sin \alpha)^x - \cos^4\alpha+\sin^4\alpha}{x - 4}$
$\displaystyle =\lim_{x\rightarrow 4} \frac{(\cos \alpha)^x -\cos^4\alpha- ((\sin \alpha)^x -\sin^4\alpha)}{x - 4}$
$\displaystyle =\lim_{x\rightarrow 4} \frac{\cos^4\alpha((\cos \alpha)^{x-4} -1)-\sin^4\alpha ((\sin \alpha)^{x-4} -1)}{x - 4}$
$\displaystyle =\cos^4\alpha.ln(\cos\alpha)-\sin^4\alpha.ln(\sin\alpha)$