If $f(x) = \begin{cases} e^x; & x \le 0 \\ |1 - x|; & x > 0 \end{cases}$,then

If $f(x) = \begin{cases} \frac{a \sin x - b x + c x^2 + x^3}{2 \log(1+x) - 2x + x^2 - \frac{2}{3}x^3} &, x \neq 0 \\ 0 &, x=0 \end{cases}$ is continuous at $x=0$,then find the relation between $a, b, c$.

If a function $f(x)$ defined by $f(x) = \begin{cases} ax^2 + bx + c, & x \leq -1 \\ 2x^2 + 4x + 1, & -1 < x < 1 \\ cx^2 + bx + a, & x \geq 1 \end{cases}$ is continuous on $\mathbb{R}$,and $\lim_{x \rightarrow \frac{3}{2}} f(x) = 14$,then find $\lim_{x \rightarrow -2} f(x)$.

Find the relationship between $a$ and $b$ so that the function $f$ defined by $f(x) = \begin{cases} ax + 1, & \text{if } x \le 3 \\ bx + 3, & \text{if } x > 3 \end{cases}$ is continuous at $x = 3$.

The function $f(x) = [x]^2 - [x^2]$ (where $[x]$ is the greatest integer less than or equal to $x$) is discontinuous at:

Let $a$ be a positive real number. If a real valued function $f(x) = \begin{cases} \frac{6^x-3^x-2^x+1}{1-\cos \left(\frac{x}{a}\right)} & \text{if } x \neq 0 \\ \log 3 \log 4 & \text{if } x=0 \end{cases}$ is continuous at $x=0$,then $a=$