A number of laws govern the computation of limits. These laws allow us to compute a limit of a more complex function based on the limits of its (simpler) components.
Basic Limit Laws:
If \(\displaystyle\lim_{x\to c}f(x)\) and \(\displaystyle\lim_{x\to c}g(x)\) exist, then:
- Constant and Linear Law: \(\displaystyle \lim_{x\to c}k = k\), \(\displaystyle \lim_{x\to c} x = c\).
- Sum Law: The limit \(\displaystyle\lim_{x\to c}f(x) + g(x)\) also exists and: \[\displaystyle\lim_{x\to c}f(x) + g(x) = \lim_{x\to c}f(x) + \lim_{x\to c}g(x)\]
- Constant Multiple Law: The limit \(\displaystyle\lim_{x\to c}k f(x)\) also exists and: \[\displaystyle\lim_{x\to c}k f(x) = k\lim_{x\to c}f(x)\]
- Product Law: The limit \(\displaystyle\lim_{x\to c}f(x)g(x)\) also exists and: \[\displaystyle\lim_{x\to c}f(x)g(x) = \lim_{x\to c}f(x)\lim_{x\to c}g(x)\]
- Quotient Law: If further \(\displaystyle\lim_{x\to c}g(x)\neq 0\), then the limit \(\displaystyle\lim_{x\to c}\frac{f(x)}{g(x)}\) also exists and: \[\displaystyle\lim_{x\to c}\frac{f(x)}{g(x)} = \frac{\displaystyle\lim_{x\to c}f(x)}{\displaystyle\lim_{x\to c}g(x)}\]
- Power Law: The limit \(\displaystyle\lim_{x\to c}f(x)^n\) also exists and: \[\displaystyle\lim_{x\to c}f(x)^n = \left(\lim_{x\to c}f(x)\right)^n\]
- Root Law: If further \(\displaystyle\lim_{x\to c}g(x)\neq 0\), then the limit \(\displaystyle\lim_{x\to c}\sqrt[n]{f(x)}\) also exists and: \[\displaystyle\lim_{x\to c}\sqrt[n]{f(x)} = \sqrt[n]{\lim_{x\to c}f(x)}\]
For example, let’s compute \(\displaystyle\lim_{x\to 1} \sqrt{x^3 + 2x}\):