Limit algorithms
Theorem 3
If \ (\ Lim F (X) = A \) , \ (\ Lim G (X) = B \) , then
- \(\lim[f(x)\pm g(x)]=\lim f(x)\pm \lim g(x)=A+B\);
- \(\lim[f(x)\cdot g(x)]=\lim f(x)\cdot \lim g(x)=A\cdot B\);
- If there \ (B \ 0 = Not \) , then $$ lim \ frac {f (x )} {g (x)} = \ frac {\ lim f (x)} {\ lim g (x)} = \ frac {A} {B }. $$
:( feel more evidence is clear, but the book above proved to be too much trouble, you can think about your own proof)
Corollary 1: If \ (\ lim f (x) \) exist, and \ (C \) is constant, then $$ \ lim [cf (x) ] = c \ lim f (x) $$.
Corollary 2: If \ (\ lim f (x) \) exist, and \ (n-\) is a positive integer, then $$ \ lim [f (x) ] ^ n = [\ lim f (x)] ^ n $ $
Theorem 4
With the number of columns \ (\ {x_n \} \ ) and \ (\ {y_n \} \) . If $$ \ lim_ {x \ to \ infty} x_n = A, \ lim_ {x \ to \ infty} y_n = B, $$ then
- \(\lim_{x\to\infty}(x_n\pm y_n)=A+B\);
- \(\lim_{x\to\infty}(x_n\cdot y_n)=A\cdot B\);
- 当 \(y_n\not=0(n=1,2,\dots)\) 且 \(B\not=0\) 时,\(\lim_{x\to\infty}\frac{x_n}{y_n}=\frac{A}{B}\).
Theorem 5
** If \ (G (X-) \ GEQ F (X) \) , and \ (\ Lim G (X) = A, \ Lim F (X) = B \) , then the \ (A \ geq B \) .