Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined as $$f(x) = a \sin\left(\frac{\pi[x]}{2}\right) + [2 - x]$$, $$a \in \mathbb{R}$$, where $$[t]$$ is the greatest integer less than or equal to $$t$$. If $$\lim_{x \to -1} f(x)$$ exists, then the value of $$\int_0^4 f(x) \, dx$$ is equal to

We are given $$f(x) = \dfrac{x}{(1+x)^{1/x}}$$ for $$x > 0$$ and $$f(0) = e$$.

To check continuity at $$x = 0$$, we evaluate $$\displaystyle\lim_{x \to 0^+} f(x)$$.

We know the standard limit $$\displaystyle\lim_{x \to 0} (1+x)^{1/x} = e$$.

As $$x \to 0^+$$, the numerator $$x \to 0$$ and the denominator $$(1+x)^{1/x} \to e$$.

Therefore $$\displaystyle\lim_{x \to 0^+} f(x) = \dfrac{0}{e} = 0$$.

But $$f(0) = e \neq 0$$. Since the limit does not equal the function value, $$f$$ is not continuous at $$x = 0$$.

The correct answer is Option D.