Limit Laws

Just as most operators, like the integral and $\frac{d}{dx}$ have their own respective laws as to what addition, multiplication, etc. mean for them, limits also have certain limit laws. These help decompose a complex limit into a simpler version.

Laws §

1. ${\lim }_{x\to a}[f(x)+g(x)]={\lim }_{x\to a}f(x)+{\lim }_{x\to a}g(x)$

Addition does not affect limits, we can ‘pull out’ the plus sign from the limit. E.g. ${\lim }_{x\to \infty }[x^2+\log (x)]={\lim }_{x\to \infty }(x^2)+{\lim }_{x\to \infty }(\log (x))$

2. ${\lim }_{x\to a}[c\times f(x)]=c\times {\lim }_{x\to a}f(x)$

Constant factors multiplication (and division) do not affect limits. E.g. ${\lim }_{x\to \infty }\left[\frac{39}{x^2}\right]=39\times {\lim }_{x\to \infty }\left(\frac{1}{x^2}\right)$

3. ${\lim }_{x\to a}[f(x)\times g(x)]={\lim }_{x\to a}f(x)\times {\lim }_{x\to a}g(x)$

4. ${\lim }_{x\to a}\left[\frac{f(x)}{g(x)}\right]=\frac{{\lim }_{x\to a}f(x)}{{\lim }_{x\to a}g(x)}if{\lim }_{x\to a}g(x)\ne 0$

5. ${\lim }_{x\to a}c=cifc\ne \infty$

6. ${\lim }_{x\to a}x=aifa\ne \infty$