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Location Engineering Building | room 332
Phone 435-797-1494
Contact Christian Bolander
Email christian.bolander@usu.edu

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Fri	10am – 4pm

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The Limit Laws

Engineering Context:

Limits provide the basis of our understanding of how systems change over time, and are formally used when taking derivatives. Additionally, numberical simulations often require applications of limits. Understanding limit laws and properties of limits can help us to better understand what limits are actually doing, and can help us to compute formal limits when necessary.

The Essentials

There are certain properties of limits that help us manipulate equations in order to find limits of composite functions. They are shown below.

For each law let $f (x)$ and $g (x)$ be defined for all $x \neq a$ over an open interval containing $a$ . Let $c$ be any contant.

Basic Limit Results

lim_{x \to a} x = a

lim_{x \to a} c = c

Sum Law

lim_{x \to a} (f (x) + g (x)) = lim_{x \to a} f (x) + lim_{x \to a} g (x)

Difference Law

lim_{x \to a} (f (x) - g (x)) = lim_{x \to a} f (x) - lim_{x \to a} g (x)

Constant Multiple Law

lim_{x \to a} c f (x) = c \cdot lim_{x \to a} f (x)

Product Law

lim_{x \to a} (f (x) \cdot g (x)) = lim_{x \to a} f (x) \cdot lim_{x \to a} g (x)

Quotient Law

lim_{x \to a} \frac{f (x)}{g (x)} = \frac{lim_{x \to a} f (x)}{lim_{x \to a} g (x)}

Power Law

lim_{x \to a} {(f (x))}^{n} = ({lim_{x \to a} f (x))}^{n}

Root Law

lim_{x \to a} \sqrt[n]{f (x)} = \sqrt[n]{lim_{x \to a} f (x)}

A Deeper Dive

In order to find soem limits, we may need to use several limit laws at the same time or even repeatedly . As we solve a more complex limit problem we need to remember to rewrite the limit in terms of other limits each time we apply a new limit law. See the example below:

lim_{x \to -3} \frac{x^{2} + 5 x + 6}{x + 2}

Apply quotient law, checking that $(-3) + 2 \neq 0$ .

lim_{x \to -3} \frac{x^{2} + 5 x + 6}{x + 2} = \frac{lim_{x \to -3} (x^{2} + 5 x + 6)}{lim_{x \to -3} (x + 2)}

Apply sum law and constant multiple law.

= \frac{lim_{x \to -3} x^{2} + 5 \cdot lim_{x \to -3} x + lim_{x \to -3} 6}{lim_{x \to -3} x + lim_{x \to -3} 2}

Apply basic limit laws and simplify.

= \frac{{(-3)}^{2} + 5 (-3) + 6}{(-3) + 2} = \frac{0}{-1} = 0

Practice

Evaluate the following limit:

lim_{x \to 5} \frac{x^{2} + 4 x}{x^{2} + 8 x + 16}

Solution: There are two ways to solve this problem.

Method 1:

Use the quotient law to get:

\frac{lim_{x \to 5} x^{2} + 4 x}{lim_{x \to 5} x^{2} + 8 x + 16}

Now take the limit of the numberator and denominator separately to get:

\frac{{(5)}^{2} + 4 (5)}{{(5)}^{2} + 8 (5) + 16} = \frac{45}{81} = \frac{5}{9}

Method 2:

Factor the numerator and denominator to get:

lim_{x \to 5} \frac{x (x + 4)}{(x + 4) (x + 4)}

When we simplify this equation, we get:

lim_{x \to 5} \frac{x}{(x + 4)} = \frac{(5)}{((5) + 4)} = \frac{5}{9}

The Limit Laws

Engineering Context:

The Essentials

A Deeper Dive

Practice

More Resources