Granica funkcji całość

[martaaa7]

a) \(\lim_{x\to 0} \frac{x}{a} \left[ \frac{b}{x} \right]\)
b) \(\lim_{x\to0 } \frac{b}{x} \left[ \frac{x}{a} \right]\)

[octahedron]

a)
\(\frac{b}{x}-1\le\[\frac{b}{x}\]\le\frac{b}{x}
\frac{x}{a}>0:
\frac{x}{a}\(\frac{b}{x}-1\)\le\frac{x}{a}\[\frac{b}{x}\]\le\frac{x}{a}\cdot\frac{b}{x}
\frac{x}{a}<0:
\frac{x}{a}\(\frac{b}{x}-1\)\ge\frac{x}{a}\[\frac{b}{x}\]\ge\frac{x}{a}\cdot\frac{b}{x}
\frac{b}{a}-\frac{x}{a}\ge\frac{x}{a}\[\frac{b}{x}\]\ge\frac{b}{a}
\lim_{x\to 0}\lim_{x\to 0}\frac{b}{a}-\frac{x}{a}=\frac{b}{a}\Rightarrow\frac{x}{a}\[\frac{b}{x}\]=\frac{b}{a}\)

[octahedron]

b)
\(0\le\frac{x}{a}<1\Rightarrow \[\frac{x}{a}\]=0
-1\le\frac{x}{a}<0\Rightarrow \[\frac{x}{a}\]=-1
a>0:
\lim_{x\to 0^-}\frac{b}{x}\[\frac{x}{a}\]=\lim_{x\to 0^-}-\frac{b}{x}=\infty
\lim_{x\to 0^+}\frac{b}{x}\[\frac{x}{a}\]=\lim_{x\to 0^+}\frac{b}{x}\cdot 0=\lim_{x\to 0^+}0=0
a<0:
\lim_{x\to 0^-}\frac{b}{x}\[\frac{x}{a}\]=\lim_{x\to 0^-}\frac{b}{x}\cdot 0=\lim_{x\to 0^-}0=0
\lim_{x\to 0^+}\frac{b}{x}\[\frac{x}{a}\]=\lim_{x\to 0^+}-\frac{b}{x}=-\infty\)