**15.11. Limit at Infinity**

**Polynomial Functions**

Finding the limit of a polynomial function at infinity is the same as finding the limit as

You can determine the limit of a function as it approaches -∞

1) When the degree is

The limit at +∞ is +∞.

The limit at -∞ is -∞.

The limit at +∞ is -∞.

The limit at -∞ is +∞.

2) When the degree is

The limit at +∞ is +∞.

The limit at -∞ is +∞.

The limit at +∞ is -∞.

The limit at -∞ is -∞.

*x*approaches**+∞**.**Similarly, the limit at negative infinity is the same as finding the limit as***x*approaches -∞. If f(x) increases as*x*increases then the limit of f(x) as*x*approaches +∞ is**+∞.**If f(x) decreases as*x*approaches -∞, then the limit of f(x) as*x*approaches -∞ is**-∞**.You can determine the limit of a function as it approaches -∞

**or****+∞**based on its degree (the highest exponent).1) When the degree is

**odd**, such as f(x) = x or f(x) = x^3**Case 1: a > 0**The limit at +∞ is +∞.

The limit at -∞ is -∞.

**Case 2: a < 0**The limit at +∞ is -∞.

The limit at -∞ is +∞.

2) When the degree is

**even**, such as f(x) = x^2 or f(x) = x^4**Case 1: a > 0**The limit at +∞ is +∞.

The limit at -∞ is +∞.

**Case 2: a < 0**The limit at +∞ is -∞.

The limit at -∞ is -∞.

**Illustrative Example**

Find the limit of f(x) = x^3 - 4x^2 + 6 as

*x*approaches -∞.This is a 3rd degree polynomial with a = 1 > 0. So the limit as

Step 1: Tap and hold the exponent key. Select "lim." You can also tap the exponent key four times.

*x*approaches -∞ is -∞.**Calculator solution**Step 1: Tap and hold the exponent key. Select "lim." You can also tap the exponent key four times.

Step 2: Enter infinity by holding the degree key and selecting ∞. You can also tap the degree key four times.

Step 3: Enter the function in parentheses.

Type in: lim [ x = -∞ ] ( 9x^3 - 4x^2 + 6 )

Type in: lim [ x = -∞ ] ( 9x^3 - 4x^2 + 6 )

**More Examples**

Find the limit at infinity for each polynomial function below.

**Calculator solution**

Type in: lim [ x = ∞ ] 3

**Calculator solution**

Type in: lim [ x = ∞ ] ( 3x^5 - 4x + 2x - 3 )

**Calculator solution**

Type in: lim [ x = - ∞ ] ( -3x^4 - 4x^3 + 2x - 3 )

**Calculator solution**

Type in: lim [ x = ∞ ] ( x^2 - x^4 )

**Rational Functions**

To find the limit at infinity of a rational function, let

1) If the degree of the numerator is equal to the degree of the denominator, the limit at infinity is a/b. In the example below, the degrees are the same ( x^3 ), so the limit at infinity is 4/2 = 2.

**ax^n**be the first term of the numerator and bx^m be the first term of the denominator.1) If the degree of the numerator is equal to the degree of the denominator, the limit at infinity is a/b. In the example below, the degrees are the same ( x^3 ), so the limit at infinity is 4/2 = 2.

2) If the degree of the numerator is less than the degree of the denominator, then the limit at infinity is 0. In the example below, the degree of the numerator is x^2 which is less than the degree of the denominator x^3, so the limit is 0.

3) If the degree of the numerator is greater than the denominator, the limit is either positive or negative infinity.

**More Examples**

Calculate the limit at infinity for each rational function below.

**Calculator solution**

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

**Calculator solution**

Type in: lim [ z = -∞ ] ( ( 4z^2 + z ^ 6 ) / ( 1 + 5z^3 ) )

**Calculator solution**

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

**Calculator solution**

Type in: lim [ x = ∞ ] ( ( 4x^3 ) / ( 5x^2 - 3 ) )

**Calculator solution**

Type in: lim [ x = ∞ ] ( ( 5x^2 - 7x + 9 ) / ( x^2 -2x - 3 ) )

**Calculator solution**

Type in: lim [ x = ∞ ] ( ( 3x^2 - 7x ) / ( x - 8 ) )

**Radical Functions**

To find the limit at infinity of a radical function, you can substitute large values of

*x*to see the behavior of the function f(x). The limit also depends on the degree of the radical and the radicand. If the radicand contains a fraction, the rules for rational expressions hold.

Illustrative ExamplesIllustrative Examples

Calculate the limit at infinity for each radical function below.

As

*x*increases, the value of sqrt(3x - 2) increases. So the limit is +∞.The domain of the radical function is the set of real numbers greater than or equal to 3/2, so the limit as

*x*approaches -∞ does not exist.The function is positive for any

*x*, so plugging in a large negative number into f(x) would result in a large positive number. The limit is +∞.**More Examples**

Calculate each limit at infinity below.

**Calculator solution**

Type in: lim [ x = -∞ ] cbrt( ( x - 3 ) / ( 5 - x ) )

**Calculator solution**

Type in: lim [ x = ∞ ] ( x + sqrt( x^2 + 2x ) )

**Calculator solution**

Type in: lim [ x = ∞ ] ( ( sqrt x - 3 ) / ( x - 9 ) )

**Calculator solution**

Type in: lim [ x = ∞ ] ( sqrt( x^6 + 3x^2 + 1 ) / ( 4x^3 + 3 ) )

**Calculator solution**

Type in: lim [ x = ∞ ] ( sqrt( x^2 + 3 ) / ( 7x + 5 ) )

**Absolute Value Functions**

The graph of an absolute value function may increase or decrease at all values of

*x*depending on whether it opens upward or downward.**Examples**

Calculate each limit at infinity if it exists.

**Calculator solution**

Type in: lim [ x = -∞ ] abs ( 2x + 3 ) - 3

**Calculator solution**

Type in: lim [ x = -∞ ] ( -abs ( 3x + 2 ) + 2 )

**Calculator solution**

Type in: lim [ x = ∞ ] (2 ( abs( x^2 - 4 ) - abs( x ) - 4 ) )

**Calculator solution**

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

**Calculator solution**

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

**Exponential Functions**

An exponential function with a base b > 1 has a graph that increases as

When 0 < b < 1, the reverse is true. The limit at positive infinity is the y-value of the asymptote while the limit at negative infinity is +∞.

*x*increases, so the limit at positive infinity is +∞. Note that an exponential function f(x) = b^x also has an asymptote. The asymptote is the limit at negative infinity.When 0 < b < 1, the reverse is true. The limit at positive infinity is the y-value of the asymptote while the limit at negative infinity is +∞.

**Illustrative Examples**Calculate each limit at infinity.

Since b = 3 > 1, the limit at infinity is +∞.

Since b = 1/2 < 1, f(x) approaches the asymptote y = 3. The limit at positive infinity is 3.

**More Examples**

Calculate each limit below if it exists.

**Calculator solution**

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

**Calculator solution**

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

**Calculator solution**

Type in: lim [ x = -∞ ] ( 2 ^( x - 2 ) + 3 )

**Calculator solution**

Type in: lim [ x = -∞ ] ( -e^( -2x ) + 3 )

**Logarithmic Functions**

A logarithmic function is the inverse of an exponential function. Since the range of an exponential function is the set of real numbers greater than zero, the domain of a logarithmic function is the set of real numbers greater than zero. Therefore evaluating limits at negative infinity is not possible. A logarithmic function also has a vertical asymptote. The expression and base of the logarithmic function determine whether it approaches negative or positive infinity.

**Illustrative Examples**Calculate each limit at infinity if it exists.

As

*x*increases, f(x) increases, so the limit is +∞.Because the domain must be positive, the limit at negative infinity is undefined.

**More Examples**

Calculate each limit at infinity if it exists.

**Calculator solution**

Type in: lim [ x = ∞ ] log ( 2x )

**Calculator solution**

Type in: lim [ x = ∞ ] log ( 2x^2 - 3 )

**Calculator solution**

Type in: lim [ x = ∞ ] ( log ( 2x^3 - 3 ) + 2 ln x )

**Calculator solution**

Type in: lim [ x = ∞ ] log [2] ( 3x )

**Calculator solution**

Type in: lim [ x = ∞ ] ( log [2] ( 3x - 5 ) + 2 )

**Trigonometric Functions**

Since most trigonometric functions are periodic (oscillating), limits at infinity do not exist except when the trigonometric expressions are part of an algebraic expression.

**Illustrative Examples**Calculate each limit at infinity below.

**More Examples**

Calculate each limit at infinity if it exists.

**Calculator solution**

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

**Calculator solution**

Type in: lim [ x = ∞ ] ( x sin ( 1 / x ) )

**Functions with No Limit at**

*x*=*c*Some functions oscillate as

*x*increases such that the limit at infinity does not exist.**Examples****Calculator solution**

Type in: lim [ x = ∞ ] ( x / cos ( -3x ) )

**Calculator solution**

Type in: lim [ x = ∞ ] cos ( 2x )

**Calculator solution**

Type in: lim [ x = -∞ ] log ( 2x - 3 )

**Calculator solution**

Type in: lim [ x = -∞ ] log [2] ( 3x )