Limits: Limit Laws

Step 1: Apply the Product of Limits Law 4.

$\underset{x\to a}{\lim }\left[f\left(x\right)g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)·\underset{x\to a}{lim}g\left(x\right)$

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

Step 2: Apply the Difference and Sum of Limit Laws 2 & 1.

Law 2: $\underset{x\to a}{\lim }\left[f\left(x\right)-g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)-\underset{x\to a}{lim}g\left(x\right)$

Law 1: $\underset{x\to a}{\lim }\left[f\left(x\right)+g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)+\underset{x\to a}{lim}g\left(x\right)$

$[\underset{x\to 2}{\lim }2x^2-\underset{x\to 2}{\lim 2}\left]·\underset{x\to 2}{[\text{lim}} x^3-\underset{x\to 2}{\text{lim}} x+\underset{x\to 2}{\text{lim}}5\right]$

Step 3: Apply the Constant Multiple Law 3.

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

$[2 \underset{x\to 2}{\lim }{x}^2-\underset{x\to 2}{\lim 2}\left]·\underset{x\to 2}{[\text{lim}} x^3-\underset{x\to 2}{\text{lim}} x+\underset{x\to 2}{\text{lim}} 5\right]$

Step 4: Apply Limit Law 9.

$\underset{x\to a}{\lim }{x}^n=a^n$

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

Step 5: Apply Limit Law 7.

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

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

Step 6: Apply Limit Law 8.

$\underset{x\to a}{\lim }x=a$

$\left[2\right(2^2)-2\left]·\right[2^3-2+5]$

Step 7: Evaluate the limit as $x\to 2$.

$[2(2^2)-2\left]·\right[2^3-2+5]=66$

$\underset{x\to 2}{\lim }\left(2x^2-2\right)\left(x^3-x+5\right)=66$

Step 1: Apply the Product of Limits Law 4.

$\underset{x\to a}{\lim }\left[f\left(x\right)g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)·\underset{x\to a}{lim}g\left(x\right)$

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

Step 2: Apply the Difference and Sum of Limit Laws 2 & 1.

Law 2: $\underset{x\to a}{\lim }\left[f\left(x\right)-g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)-\underset{x\to a}{lim}g\left(x\right)$

Law 1: $\underset{x\to a}{\lim }\left[f\left(x\right)+g\left(x\right)\right]=\underset{x\to a}{lim}f\left(x\right)+\underset{x\to a}{lim}g\left(x\right)$

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

Step 3: Apply the Constant Multiple Law 3.

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

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

Step 4: Apply Limit Law 7.

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

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

Step 5: Apply Limit Law 9.

$\underset{x\to a}{\lim }{x}^n=a^n$

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

Step 6: Apply Limit Law 11.

$\underset{x\to a}{\lim }\sqrt[n]{f\left(x\right)}=\sqrt[n]{\underset{x\to a}{\lim }f\left(x\right)}$

$[3-5 (8^2)+8^3 \left]·\right[2+\sqrt[3]{8}]$

Step 7: Evaluate the limit as $x\to 2$.

$[3-5 (8^2)+8^3 \left]·\right[2+\sqrt[3]{8}]=780$

$\underset{x\to 8}{\lim }\left(3-5x^2+x^3\right)\left(2+\sqrt[3]{x}\right)=780$