<s>[INST] Evaluating $\lim_\limits{x\to 1 }\bigl( (2^x x + 1)/(3^x x)\bigr)^{\tan(\pi x/2)}$ I have to calculate limit 
$$\lim_{x\to 1 } \left(\frac{2^x x + 1}{3^x x}\right)^{\tan(\frac{\pi x}{2})}.$$
I know $\tan(\frac{\pi x}{2})$ is undefined in $x = 1$, but can I just put $x = 1$ into $\frac{x\cdot 2^x + 1}{x\cdot3^x}$ and get 
$$\lim_{x\to 1 } (1)^{\tan(\frac{\pi x}{2})} = 1.$$ 
Is the answer $1$ correct?
It's forbidden to use L'Hôpital's rule.
 [/INST] Let $$y = \left( \frac{x 2^x + 1}{x 3^x}\right)^{\tan\left(\frac{\pi x}{2}\right)}$$
Then $$\ln(y) = \tan\left(\frac{\pi x}{2}\right)\ln\left( \frac{x 2^x + 1}{x 3^x}\right) = \frac{\ln\left( \frac{x 2^x + 1}{x 3^x}\right)}{\frac{1}{\tan\left(\frac{\pi x}{2}\right)}}$$
Apply L'Hôpital's rule to obtain your answer, it is still a lot of work. Maybe there is a faster way to go about this problem.
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