Properties of Limit Function

Finding a Limit from the Limits of Its Parts

The limit rules calculate the limit of a sum, difference, product, or quotient from the limits of its parts. Check that the component limits exist. For a quotient, the denominator limit must also be nonzero.

Basic Properties of Limits

Two rules cover the simplest cases, the limit of a constant function and the limit of the identity function. Both are read directly from the graph, and every other rule is built on top of them.

Constant Property

The simplest property is the limit of a constant function. If $k$ is a constant, then:

A constant does not change as $x$ approaches $c$, so its limit is the constant itself.

Identity Property

For the identity function, the following holds:

When $x$ approaches $c$, the value of function $f(x) = x$ also approaches $c$.

Arithmetic Operation Properties

Suppose $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$ where $L$ and $M$ are real numbers. The rules then move the limit through a sum, a product, a constant factor, and a quotient, provided the denominator does not approach zero.

Addition and Subtraction Properties

The limit of the sum or difference of two functions equals the sum or difference of the limits of each function:

A compound limit becomes several simpler limits that can be evaluated separately.

Multiplication Property

The limit of the product of two functions equals the product of the limits of each function:

Multiplication by Constant Property

A constant can be factored out from the limit sign:

Division Property

The limit of the quotient of two functions equals the quotient of the limits of each function, provided the limit of the denominator is not zero:

The quotient rule applies when $M \neq 0$.

Power and Root Properties

A power or a root passes through the limit when the result stays inside the domain. That domain condition keeps the root defined, so a negative value under an even root stops the rule from applying.

Power Property

The limit of a function raised to a power equals the power of the limit of the function:

In this rule, $n$ is a positive integer.

Root Property

The limit of the root of a function equals the root of the limit of the function:

Domain condition: The root must be defined for inputs near the point being approached.

• If $n$ is odd: the real root is defined for every real value of $L$
• If $n$ is even: the function under the root must remain nonnegative near the point, and $L \geq 0$

Examples of Applying Limit Properties

Each example applies one rule at a time and names it, so you can follow the same order on a similar expression. The last example shows what changes when a root is involved.

Limit of a Polynomial

Calculate $\lim_{x \to 2} (3x^2 + 5x - 1)$.

Solution:

Using limit properties:

Limit of a Rational Function

Calculate $\lim_{x \to 4} \frac{x\sqrt{x}}{x^2 + 3}$.

Solution:

Using division and multiplication properties:

Now we substitute the value $x = 4$:

In decimal form: $\frac{8}{19} \approx 0.421$

Limit of a Radical Function

Calculate $\lim_{x \to 0} \sqrt{x^2 - 3x + 2}$.

Solution:

Using the root property (since $n = 2$ is even, we need to ensure the result inside the root is not negative):

Calculate the limit inside the root first:

Since $2 > 0$, we can use the root property:

Exercises

Each problem asks for a limit that uses one or more of the rules above. Write down the rule used at every step.

1. Calculate $\lim_{x \to 3} (2x^2 - 4x + 1)$

2. Calculate $\lim_{x \to 1} \frac{3x + 2}{x^2 + 1}$

3. Calculate $\lim_{x \to 4} \sqrt{x + 5}$

4. Calculate $\lim_{x \to 2} (x + 1)^3$

5. Calculate $\lim_{x \to 0} \frac{5x^2 + 3x}{2x + 1}$

Worked Solutions

1. Solution:

   Using addition and multiplication by constant properties:

   $$\begin{aligned} &\lim_{x \to 3} (2x^2 - 4x + 1) \\ &= 2\lim_{x \to 3} x^2 \\ &\quad{}- 4\lim_{x \to 3} x + \lim_{x \to 3} 1 \end{aligned}$$

   Substitute $x = 3$:

   $$\begin{aligned} &= 2(3)^2 - 4(3) + 1 \\ &= 2(9) - 12 + 1 \\ &= 18 - 12 + 1 \\ &= 7 \end{aligned}$$

2. Solution:

   Using the division property:

   $$\lim_{x \to 1} \frac{3x + 2}{x^2 + 1} = \frac{\lim_{x \to 1} (3x + 2)}{\lim_{x \to 1} (x^2 + 1)}$$

   $$= \frac{3(1) + 2}{1^2 + 1} = \frac{5}{2}$$

   In decimal form: $\frac{5}{2} = 2.5$

3. Solution:

   Using the root property:

   $$\lim_{x \to 4} \sqrt{x + 5} = \sqrt{\lim_{x \to 4} (x + 5)}$$

   $$= \sqrt{4 + 5} = \sqrt{9} = 3$$

4. Solution:

   Using the power property:

   $$\lim_{x \to 2} (x + 1)^3 = \left[\lim_{x \to 2} (x + 1)\right]^3$$

   $$= (2 + 1)^3 = 3^3 = 27$$

5. Solution:

   Using the division property:

   $$\lim_{x \to 0} \frac{5x^2 + 3x}{2x + 1} = \frac{\lim_{x \to 0} (5x^2 + 3x)}{\lim_{x \to 0} (2x + 1)}$$

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