Properties of limits
| Name | Rule | Notes |
|---|---|---|
| Constant function | \(\lim\limits_{x\to a}f(z)=f(z)\) | \(f(z)\) does not depend on \(x\) |
| Constant multiple of a function | \(\lim\limits_{x\to a}[kf(x)]=k\lim\limits_{x\to a}f(x)\) | A multiplied constant can be taken out of the limit |
| Sum/difference of functions | \(\lim\limits_{x\to a}[f(x)\pm g(x)]=\lim\limits_{x\to a}f(x)\pm\lim\limits_{x\to a}g(x)\) | \(x\to a\) remains the same |
| Product of functions | \(\lim\limits_{x\to a}[f(x)\cdot g(x)]=\lim\limits_{x\to a}f(x)\cdot\lim\limits_{x\to a}g(x)\) | \(x\to a\) remains the same |
| Quotient of functions | \(\lim\limits_{x\to a}\dfrac{f(x)}{g(x)}=\dfrac{\lim\limits_{x\to a}f(x)}{\lim\limits_{x\to a}g(x)}\) | \(x\to a\) remains the same Provided \(\lim\limits_{x\to a}g(x)\neq 0\) |
Power of a function | \(\lim\limits_{x\to a}\left[f(x)\right]^n=\left[\lim\limits_{x\to a}f(x)\right]^n\) | \(x\to a\) remains the same Provided \(\lim\limits_{x\to a}f(x)\) exists |