Given a function: $$ f(x)= \lim_{t\to+\infty} (1+x)\tan(xt) $$ Study its continuity and sketch a graph.
This problem comes along with several similar ones, one of which was to study the properties of the function: $$ g(x) = \lim_{t\to+\infty}\frac{\ln(1+e^{xt})}{\ln(1+e^t)} $$
After some algebraic manipulations, one can conclude that the function is $g(x) = 0$ for $x \le 0$ and $g(x) = x$ for $x > 0$: $$ \lim_{t\to+\infty}\frac{\ln(1+e^{xt})}{\ln(1+e^t)} \stackrel{x > 0}{=} \\ \lim_{t\to+\infty}\frac{\ln(e^{xt})+\ln\left(1+{1\over e^{xt}}\right)}{\ln(e^t)+\ln\left(1+{1\over e^t}\right)} \sim \frac{xt}{t} = x $$ For $x \le 0$: $$ \lim_{t\to+\infty}\frac{\ln(1+e^{xt})}{\ln(1+e^t)} \stackrel{x \le 0}{=}\\ \lim_{t\to+\infty}\frac{\ln\left(1+{1\over e^{|x|t}}\right)}{\ln(1+e^t)} = 0 $$
A similar approach has worked for several problems of a similar kind, but the one in the question section is trickier.
Obviously for $x = 0$: $$ \lim_{t\to+\infty}(1+x)\tan(xt) = 0 $$
But for other values of $x$ I couldn't study its behavior because of $\tan tx$ doesn't even have a limit for $x\ne \pi n$ and $t\to +\infty$. I've referred to desmos in order to get more insights on it, but it became even more convoluted.
The question is how to study the continuity of the given function? And what its graph looks like?
Note that $f$ is the null map from $\{-1,0\}$ into $\mathbb R$. And so $f$ is continuous.