Limit Laws

Basic limits

$\underset{x\to \omega }{\lim }𝒸=𝒸\underset{x\to \omega }{\lim }x=\omega$

Sum

$\underset{x\to \omega }{\lim }\left(f\left(x\right)+g\left(x\right)\right)=\underset{x\to \omega }{\lim }f\left(x\right)+\underset{x\to \omega }{\lim }g\left(x\right)$

Difference

$\underset{x\to \omega }{\lim }\left(f\left(x\right)-g\left(x\right)\right)=\underset{x\to \omega }{\lim }f\left(x\right)-\underset{x\to \omega }{\lim }g\left(x\right)$

Constant multiple

$\underset{x\to \omega }{\lim }𝒸f\left(x\right)=𝒸\underset{x\to \omega }{\lim }f\left(x\right)$

Product

$\underset{x\to \omega }{\lim }\left(f\left(x\right)g\left(x\right)\right)=\left(\underset{x\to \omega }{\lim }f\left(x\right)\right)\left(\underset{x\to \omega }{\lim }g\left(x\right)\right)$

Division

Assuming $\underset{x\to \omega }{\lim }g\left(x\right)\ne 0$

$\underset{x\to \omega }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to \omega }{\lim }f\left(x\right)}{\underset{x\to \omega }{\lim }g\left(x\right)}$

Power

Assuming $n\in {\mathbb{Z}}^{+}$

$\underset{x\to \omega }{\lim }{\left(f\left(x\right)\right)}^n={\left(\underset{x\to \omega }{\lim }f\left(x\right)\right)}^n$

Root

Assuming $n\in {\mathbb{Z}}^{+}$ is even.

Assuming $n\in {\mathbb{Z}}^{+}$ is odd and $\underset{x\to \omega }{\lim }f\left(x\right)>0$.

Polynomials and rational functions

Let $p\left(x\right),q\left(x\right)$ be two polynomials

If $q\left(\omega \right)\ne 0$ then

Limit rules also apply to:

• two sided limits,
• one sided limits,
• limits at positive infinity,
• limits at negative infinity.

Examples

• $\underset{x\to 5}{lim}2$
• $\underset{x\to -2}{lim}x$
• $\underset{x\to \mathrm{𝒷}}{lim}(x-2)$

• $\underset{x\to 1}{lim}(4x^3+3x-1)\sqrt{x+8}$
• $\underset{x\to 𝒷}{lim}\frac{5x^4+x^3+x^2-7}{x^3+3}$

• $\underset{x\to 3^{-}}{lim}\frac{{(x-3)}^2}{x^2-9}$
• $\underset{x\to 3^{-}}{lim}\frac{x^2-9}{{(x-3)}^2}$
• $\underset{x\to 3^{+}}{lim}\frac{x^2-9}{{(x-3)}^2}$

• try $\underset{x\to 0}{lim}(x^{2}sin\left(\frac{1}{x}\right))$
• try $\underset{x\to 0}{lim}(\left|x\right|\left(\frac{1+cos\left(\frac{1}{x}\right)}{2}\right)-x^2\left(\frac{1-cos\left(\frac{1}{x}\right)}{2}\right))$

• try $\underset{x\to 0^{+}}{lim}sinx$
• try $\underset{x\to 0^{-}}{lim}sinx$
• $\underset{x\to 0}{lim}(1-cosx)$
• $\underset{x\to 0}{lim}\frac{sinx}{x}$, see its graph
• $\underset{x\to 0}{lim}\frac{(1-cosx)}{x}$

Squeeze Theorem