**15.5. Limit of a Rational Function**

**Limit of a Rational Function**

A rational function may have a restricted value at

*x*=

*c*such that finding the limit is not straightforward. The rules are listed as follows:

1) Determine the restricted values for the domain of the function.

To find these values, set the denominator to 0 and find the roots of the resulting equation.

**Example**

f(x) = 3/(x - 4)

When x = 4, the function is undefined, implying that the graph has an infinite discontinuity such that the limit does not exist as

*x*approaches 4. The line x=4 is a vertical asymptote of the graph.

2) If

*c*is a restricted value, the limit may or may not exist.To determine whether it exists, simplify the rational expression. The limit exists if the simplified form is no longer a rational expression or the denominator is no longer zero when

*x*=

*c*.

3) Plug in

*c*after you simplify the expression. If the denominator is still zero, find the one-sided limits of the function by plugging in values that are close to

*c*from the left and right. If the one-sided limits are equal, the limit equals the left- and right-hand limits. Otherwise the limit does not exist.

4) If

*c*is not in the restricted domain, plug it into the rational expression as you usually would.

**Finding the Limit Through Direct Substitution**

In each rational function below, the value of

*c*is not in the restricted domain so the limit can be found by plugging in the value of

*c*.

**Examples**

Find the indicated limit.

**Calculator solution**

Type in: lim [ x = -3 ] ( ( 2x ) / (3x - 1 ) )

**Calculator solution**

Type in: lim [ x = 2 + ] ( ( 3 - x ) / ( 3x + 1 ) )

**Calculator solution**

Type in: lim [ x = 1000000 ] ( 1 / ( x - 1 )^2 )

**Calculator solution**

Type in: lim [ x = 2 ] ( 1 / ( x - 1 )^2 )

**Limit at a Restricted Value of X**

In each rational function below, the value of

*c*is a restricted value of the function's domain. There are three cases that could happen:

**Case 1: The Limit Exists**

The limit exists at the restricted value if the original rational function can be simplified to cancel out the denominator. At

*x*=

*c*, the graph has a

**hole**.

**Illustrative Example**

Consider the function f(x) = (x^3 - 4x^2 + x + 6) / (x - 2) and find the limit as

*x*approaches 2. The function is undefined at x = 2, but the expression can be simplified to f(x) = x^2 - 2x - 3. Because the denominator cancels out, the limit as

*x*approaches 2

*exists*.

Note that finding the limit of f(x) = (x^3 - 4x^2 + x + 6) / (x - 2) gives the same result as finding the limit of f(x) = x^2 - 2x - 3 as shown below.

The graph below shows that the graph still has a hole at x = 2.

**More Examples**

Evaluate the limit of each rational function as

*x*approaches*c*where*c*is a restricted value of the domain.**Calculator solution**

Type in: lim [ x = 2 ] ( ( x^2 + 4x - 12 ) / ( x^2 - 2x ) )

**Calculator solution**

Type in: lim [ t = 5 ] ( ( t^3 - 6t^2 + 25 ) / ( t - 5 ) )

**Calculator solution**

Type in: lim [ x = 1 ] ( ( 2 - 2x^2 ) / ( x - 1 ) )

**Calculator solution**

Type in: lim [ x = -6 ] ( ( ( 2x + 8 ) / ( x^2 - 12 ) - 1 / x ) / ( x + 6 ) )

**Calculator solution**

Type in: lim [ h = 0 ] ( ( 2 ( -3 + h )^2 - 18 ) / h )

**Calculator solution**

Type in: lim [ x = 0 ] ( ( ( 1 / ( 3 + x ) - 1 / ( 3 - x ) ) ) / x )

**Case 2: Limit to Infinity**

The limit may approach infinity if the denominator is still zero at x = c. As a result, the graph of the function continues increasing or decreasing as

*x*approaches

*c*from the left and right. At this value of

*x*, the graph has an

**infinite discontinuity**. The limit exists if the left- and right-hand limits increase or decrease in the same direction.

**Illustrative Example**

Consider the function f(x) = 1/x^2 and find the limit as

*x*approaches 0. Because the denominator is zero when we plug in

*x*= 0, we have to evaluate the function from the left and right. The left- and right-hand limits both approach positive infinity, so the limit of f(x) as

*x*approaches 0 is ∞.

**More Examples**

**Calculator solution**

Type in: lim [ x = 0 ] ( 6 / x^2 )

**Calculator solution**

Type in: lim [ x = 1 ] ( 1 / ( x - 1 )^2 )

**Calculator solution**

Type in: lim [ x = 1 ] ( ( x - 2 ) / ( x - 1 )^2 )

**Calculator solution**

Type in: lim [ x = 0 ] ( ( 6 - 3x ) / ( 2x^2 ) )

**Calculator solution**

Type in: lim [ x = - 4 ] ( x / ( 2x + 8 )^2 )

**Case 3: The Limit Does Not Exist (DNE)**

When the denominator is zero at

*x*=

*c*, x = c is a vertical asymptote of the graph. The function either continues increasing or decreasing as

*x*approaches

*c*from the left and right. If the left- and right-hand limits are in different directions, the limit

**does not exist**.

**Illustrative Example**

Consider the limit of f(x) = 1/x as

*x*approaches 0. The function is undefined at x = 0, as shown below. Because the left-hand limit is negative infinity while the right-hand limit is positive infinity, the limit as

*x*approaches 0 does not exist.

**More Examples**

**Calculator solution**

Type in: lim [ x = -2 ] ( -4 / ( x + 2 ) )

**Calculator solution**

Type in: lim [ t = 0 ] ( 1 / t - 1 / ( t^2 + 1 ) )

**Calculator solution**

Type in: lim [ x = 4 ] ( 3 / ( 4 - x )^3 )

**Calculator solution**

Type in: lim [ x = 2 ] ( 1 / ( x - 2 ) )

**Calculator solution**

Type in: lim [ x = 4 ] ( ( x^2 - 3x - 10 ) / ( 3x - 12 ) )