I have limit:
$$\lim_{x\to0}\left (\frac{4^{\tan(x)}+ \cos(x)}{2}\right)^{\cot(x)}$$
I tried to use the natural log:
$$\lim_{x\to 0} e^{\dfrac{\ln\left(\dfrac{4^{\tan(x)}+ \cos(x)}{2}\right)}{{\tan(x)}}}$$
But I am stuck from here, I tried multiple approaches but could not find the right result which should be $2$
How should I approach this limit?
Best Answer
Use L'Hospital's rule for the exponent ln(...) only. According to this rule, you differentiate the numerator and denominator. Doing so, you get (for the exponent): $$\lim_{x\rightarrow0} \frac{\frac{ (4^{\tan(x)}ln(4)-\sin(x))/2 }{(4^{\tan(x)}+\cos(x))/2}}{\sec^2x} = \frac{\frac{ (4^0ln(4)-0)/2 }{(4^0+1)/2}}{1} = ln(2).$$ Since the limiting value of the exponent is ln(2), the limiting value of your expression is 2.