# Limits: Limit Laws

Graphs and tables can be used to guess the values of limits but these are just estimates and these methods have inherent problems. A better method is to use the following properties of limits called Limit Laws.

LIMIT LAWS:

Assume the following limits exist and c is a constant.

$\underset{x\to a}{\text{lim}}f\left(x\right)$     $\underset{x\to a}{\text{lim}}g\left(x\right)$

1. $\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)+g\left(x\right)\right]=\underset{x\to a}{\text{lim}}f\left(x\right)+\underset{x\to a}{\text{lim}}g\left(x\right)$     Sum of Limits

2. $\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)-g\left(x\right)\right]=\underset{x\to a}{\text{lim}}f\left(x\right)-\underset{x\to a}{\text{lim}}g\left(x\right)$     Difference of Limits

3. $\underset{x\to a}{\mathrm{lim}}\left[cf\left(x\right)\right]=c\underset{x\to a}{\text{lim}}f\left(x\right)$    Constant Multiple

4. $\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)g\left(x\right)\right]=\underset{x\to a}{\text{lim}}f\left(x\right)·\underset{x\to a}{\text{lim}}g\left(x\right)$Product of Limits

5.    Quotient of Limits

6. $\underset{x\to a}{\mathrm{lim}}\left[f{\left(x\right)}^{n}\right]={\left[\underset{x\to a}{\text{lim}}f\left(x\right)\right]}^{n}$where n is a positive integer

7. $\underset{x\to a}{\mathrm{lim}}c=c$

8. $\underset{x\to a}{\mathrm{lim}}x=a$

9. $\underset{x\to a}{\mathrm{lim}}{x}^{n}={a}^{n}$where n is a positive integer

10. $\underset{x\to a}{\mathrm{lim}}\sqrt[n]{x}=\sqrt[n]{a}$   where n is a positive integer and if n is even we assume a > 0

11. $\underset{x\to a}{\mathrm{lim}}\sqrt[n]{f\left(x\right)}=\sqrt[n]{\underset{x\to a}{\mathrm{lim}}f\left(x\right)}$     where n is a positive integer and if n is even, we assume that $\underset{x\to a}{\mathrm{lim}}f\left(x\right)>0$

Let's use these laws to calculate the limits in a couple examples.

Example 1: Evaluate the following limit and justify each step.

$\underset{x\to 2}{\mathrm{lim}}\left(2{x}^{2}-2\right)\left({x}^{3}-x+5\right)$

 Step 1: Apply the Product of Limits Law 4. $\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)·\underset{x\to a}{lim}g\left(x\right)$ $\underset{x\to 2}{\mathrm{lim}}\left(2{x}^{2}-2\right)·\underset{x\to 2}{\text{lim}}\left({x}^{3}-x+5\right)$ Step 2: Apply the Difference and Sum of Limit Laws 2 & 1. Law 2: $\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)-g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)-\underset{x\to a}{lim}g\left(x\right)$ Law 1: $\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)+g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)+\underset{x\to a}{lim}g\left(x\right)$ Step 3: Apply the Constant Multiple Law 3. $\underset{x\to a}{\mathrm{lim}}\left[cf\left(x\right)\right]=c\underset{x\to a}{lim}f\left(x\right)$ Step 4: Apply Limit Law 9. $\underset{x\to a}{\mathrm{lim}}{x}^{n}={a}^{n}$ Step 5: Apply Limit Law 7. $\underset{x\to a}{\mathrm{lim}}c=c$ Step 6: Apply Limit Law 8. $\underset{x\to a}{\mathrm{lim}}x=a$ $\left[2\left({2}^{2}\right)-2\right]·\left[{2}^{3}-{2}+5\right]$ Step 7: Evaluate the limit as $x\to 2$. $\left[2\left({2}^{2}\right)-2\right]·\left[{2}^{3}-2+5\right]=66$ $\underset{x\to 2}{\mathrm{lim}}\left(2{x}^{2}-2\right)\left({x}^{3}-x+5\right)=66$
Example 2: Evaluate the following limit and justify each step.

$\underset{x\to 8}{\mathrm{lim}}\left(3-5{x}^{2}+{x}^{3}\right)\left(2+\sqrt[3]{x}\right)$

 Step 1: Apply the Product of Limits Law 4. $\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)·\underset{x\to a}{lim}g\left(x\right)$ $\underset{x\to 8}{\mathrm{lim}}\left(3-5{x}^{2}+{x}^{3}\right)·\underset{x\to 8}{\text{lim}}\left(2+\sqrt[3]{x}\right)$ Step 2: Apply the Difference and Sum of Limit Laws 2 & 1. Law 2: $\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)-g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)-\underset{x\to a}{lim}g\left(x\right)$ Law 1: $\underset{x\to a}{\mathrm{lim}}\left[f\left(x\right)+g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)+\underset{x\to a}{lim}g\left(x\right)$ Step 3: Apply the Constant Multiple Law 3. $\underset{x\to a}{\mathrm{lim}}\left[cf\left(x\right)\right]=c\underset{x\to a}{lim}f\left(x\right)$ Step 4: Apply Limit Law 7. $\underset{x\to a}{\mathrm{lim}}c=c$ Step 5: Apply Limit Law 9. $\underset{x\to a}{\mathrm{lim}}{x}^{n}={a}^{n}$ Step 6: Apply Limit Law 11. $\underset{x\to a}{\mathrm{lim}}\sqrt[n]{f\left(x\right)}=\sqrt[n]{\underset{x\to a}{\mathrm{lim}}f\left(x\right)}$ Step 7: Evaluate the limit as $x\to 2$. $\underset{x\to 8}{\mathrm{lim}}\left(3-5{x}^{2}+{x}^{3}\right)\left(2+\sqrt[3]{x}\right)=780$

 Related Links: Math algebra General Differentiation Rules Exponential Differentiation Rules Calculus Topics

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