Własności granic funkcji

Jeżeli $\displaystyle\lim_{x\to x_0}f(x)=a$\displaystyle\lim_{x\to x_0}f(x)=a oraz $\displaystyle\lim_{x\to x_0}g(x)=b$\displaystyle\lim_{x\to x_0}g(x)=b, gdzie $a,b,\in\mathbb{R}$a,b,\in\mathbb{R}, to:

• $\displaystyle\lim_{x\to x_0}(c\cdot f(x))=c\cdot \lim_{x\to x_0} f(x)=c\cdot a,\quad \text{ gdzie } c\in\mathbb{R}$\displaystyle\lim_{x\to x_0}(c\cdot f(x))=c\cdot \lim_{x\to x_0} f(x)=c\cdot a,\quad \text{ gdzie } c\in\mathbb{R}

• $\displaystyle\lim_{x\to x_0}(f(x)+g(x))=\lim_{x\to x_0}f(x) + \lim_{x\to x_0}g(x)=a+b$\displaystyle\lim_{x\to x_0}(f(x)+g(x))=\lim_{x\to x_0}f(x) + \lim_{x\to x_0}g(x)=a+b

• $\displaystyle\lim_{x\to x_0}(f(x)-g(x))=\lim_{x\to x_0}f(x) -\lim_{x\to x_0}g(x)=a-b$\displaystyle\lim_{x\to x_0}(f(x)-g(x))=\lim_{x\to x_0}f(x) -\lim_{x\to x_0}g(x)=a-b

• $\displaystyle\lim_{x\to x_0}(f(x)\cdot g(x))=\lim_{x\to x_0}f(x) \cdot\lim_{x\to x_0}g(x)=a\cdot b$\displaystyle\lim_{x\to x_0}(f(x)\cdot g(x))=\lim_{x\to x_0}f(x) \cdot\lim_{x\to x_0}g(x)=a\cdot b

• $\displaystyle\lim_{x\to x_0}\frac{f(x)}{g(x)}=\frac{\displaystyle\lim_{x\to x_0}f(x)}{\displaystyle\lim_{x\to x_0}g(x)} =\frac{a}{b},\quad \text{ gdzie } b\neq0$\displaystyle\lim_{x\to x_0}\frac{f(x)}{g(x)}=\frac{\displaystyle\lim_{x\to x_0}f(x)}{\displaystyle\lim_{x\to x_0}g(x)} =\frac{a}{b},\quad \text{ gdzie } b\neq0