Infinite Limits, Limits at Infinity, and Asymptotes✅

Infinite Limits, Limits at Infinity, and Asymptotes

Infinite Limits

Consider the behavior of the function f(x) = \frac{1}{x^2} as x approaches 0.
As x takes values closer to 0 (e.g., 0.1, 0.01, 0.001), the values of f(x) increase (e.g., 100, 10,000, 1,000,000).
The closer x gets to 0, the larger f(x) becomes.
For any large value of f(x), there exists an x very close to 0 such that f(x) equals that large value.
As x approaches 0, f(x) grows without bound.
Strictly speaking, \lim_{x \to 0} \frac{1}{x^2} does not exist because f(x) becomes infinitely large.
Shorthand notation: \lim_{x \to 0} \frac{1}{x^2} = \infty. This means that as x approaches 0, the function \frac{1}{x^2} grows larger and does not tend to any specific number.
More generally, the value of a limit as x approaches a is referred to as infinity if an arbitrarily large positive value is approached or negative infinity if an arbitrarily large negative value is approached.

Example: f(x) = \frac{1}{x - a}

Consider the behavior of f(x) as x approaches a, where a is a real number.
The graph of f(x) has a vertical asymptote at x = a.
The function resembles the graph of \frac{1}{x} translated horizontally by a units.
\lim_{x \to a^-} f(x) = -\infty. As x approaches a from the left, f(x) decreases unboundedly, tending to negative infinity.
\lim_{x \to a^+} f(x) = \infty. As x approaches a from the right, f(x) increases unboundedly, tending to infinity.
These limits do not exist in the traditional sense, meaning they are not real numbers.

Example: g(x) = \frac{x + 1}{x - 1}

Analyze the behavior of the graph of g(x) as x approaches a value a, considering both left-hand and right-hand limits.
Rewrite g(x) as g(x) = 1 + \frac{2}{x - 1} to understand its graph more easily.
The graph of g(x) has a vertical asymptote at x = 1.
\lim_{x \to 1^-} g(x) = -\infty. As x approaches 1 from the left, g(x) decreases unboundedly.
\lim_{x \to 1^+} g(x) = \infty. As x approaches 1 from the right, g(x) increases unboundedly.
These limits do not exist.

Limits at Infinity

In engineering, it is important to consider limits as x approaches infinity, providing information about the limiting behavior of a function at very large values.
*Example:
f(x) = \frac{x^2 - 1}{x^2 + 1}
Consider values of x that grow unboundedly (e.g., 10, 100, 1000, …).
As x increases, the values of f(x) approach 1.
The function is even, meaning f(x) = f(-x).
As x becomes very large (positive or negative), f(x) approaches 1.
\lim_{x \to \infty} \frac{x^2 - 1}{x^2 + 1} = 1.
\lim_{x \to -\infty} \frac{x^2 - 1}{x^2 + 1} = 1.
The graph of the function has a horizontal asymptote at y = 1.

Visualizing Infinity

In the Cartesian plane \mathbb{R}^2, the positive x-direction represents positive infinity, the negative x-direction represents negative infinity, and similarly for the y-direction.

Caution: Arithmetic with Infinity

It is incorrect to directly substitute infinity into an expression and perform arithmetic operations.
For example, \frac{\infty}{\infty} is not equal to 1.
Arithmetic with infinity is undefined in this unit.

General Definition of a Limit at Infinity

Let f be a function defined on the interval (a, \infty).
\lim_{x \to \infty} f(x) = L if f(x) can be made arbitrarily close to L by taking x sufficiently large.
For example, \lim_{x \to \infty} \frac{1}{x} = 0. As x becomes larger, \frac{1}{x} approaches 0.
Similarly, \lim_{x \to -\infty} \frac{1}{x} = 0.
Using limit laws, \lim_{x \to \infty} \frac{1}{x^n} = 0 for any positive integer n.

Calculating Limits at Infinity Algebraically

The limit \lim{x \to \infty} \frac{x^2 - 1}{x^2 + 1} = 1 can be calculated algebraically using the limit \lim{x \to \infty} \frac{1}{x} = 0.
Factor out x^2 from the numerator and denominator:
\lim{x \to \infty} \frac{x^2 - 1}{x^2 + 1} = \lim{x \to \infty} \frac{x^2(1 - \frac{1}{x^2})}{x^2(1 + \frac{1}{x^2})}
Cancel the x^2 terms:
\lim_{x \to \infty} \frac{1 - \frac{1}{x^2}}{1 + \frac{1}{x^2}}
Apply limit properties:
\lim{x \to \infty} \frac{1 - \frac{1}{x^2}}{1 + \frac{1}{x^2}} = \frac{\lim{x \to \infty} 1 - \lim{x \to \infty} \frac{1}{x^2}}{\lim{x \to \infty} 1 + \lim_{x \to \infty} \frac{1}{x^2}} = \frac{1 - 0}{1 + 0} = 1

More Examples

*Example:
\lim_{x \to \infty} \frac{1 - 2x^2}{x^2 + 1}

Multiply and divide by the highest power of x in the expression (in this case, x^2).
\lim{x \to \infty} \frac{\frac{1}{x^2} - 2}{1 + \frac{1}{x^2}} = \frac{0 - 2}{1 + 0} = -2 *Example: \lim{x \to \infty} \frac{\sqrt{1 + x^2}}{x}
Multiply and divide by x.
\lim{x \to \infty} \frac{\sqrt{1 + x^2}}{x} = \lim{x \to \infty} \sqrt{\frac{1 + x^2}{x^2}} = \lim_{x \to \infty} \sqrt{\frac{1}{x^2} + 1} = \sqrt{0 + 1} = 1
The square root function is continuous, allowing the limit to be taken inside the square root.

Homework

Calculate \lim_{x \to -\infty} \frac{1 - 2x^2}{x^2 + 1}.

Limits of Rational Functions

A rational function is a fraction of two polynomials.

Examples:

Example 1:

*\lim_{x \to \infty} \frac{x^2 + 1}{3x^2 + x}

The degree of the numerator and denominator are both 2.
Factor out x^2:
\lim{x \to \infty} \frac{x^2(1 + \frac{1}{x^2})}{x^2(3 + \frac{1}{x})} = \frac{\lim{x \to \infty} 1 + \lim{x \to \infty} \frac{1}{x^2}}{\lim{x \to \infty} 3 + \lim_{x \to \infty} \frac{1}{x}} = \frac{1 + 0}{3 + 0} = \frac{1}{3}
When the degrees of the numerator and denominator are equal, the limit as x approaches infinity is the ratio of the coefficients of the highest power of x.

Example 2:

*\lim{x \to \infty} \frac{an x^n + a{n-1} x^{n-1} + … + a1 x + a0}{bn x^n + b{n-1} x^{n-1} + … + b1 x + b0} = \frac{an}{b_n}

Example 3:

*\lim_{x \to \infty} \frac{x^2 + x + 1}{x^3 + x}

The degree of the denominator (3) is greater than the degree of the numerator (2).
\lim{x \to \infty} \frac{x^2 + x + 1}{x^3 + x} = \lim{x \to \infty} \frac{\frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3}}{1 + \frac{1}{x^2}} = \frac{0 + 0 + 0}{1 + 0} = 0
When the degree of the polynomial in the denominator is greater than the degree of the polynomial in the numerator, the limit is always 0.

*\lim{x \to \infty} \frac{an x^n + … + a0}{bm x^m + … + b_0} = 0 \text{ if } m > n

Example 4:

*\lim{x \to \infty} \frac{an x^n + … + a0}{bm x^m + … + b_0} = \infty \text{ if } n > m

Limits Involving Exponential Functions

The exponential function increases faster than any integer power of x.
\lim_{x \to \infty} \frac{x^n}{e^x} = 0 for any positive integer n.

*Graphically, the exponential e^x is always above any polynomial function.

More Limits Involving Exponential Functions

*\lim_{x \to -\infty} \frac{|x|^n}{e^{-x}} = 0

*Example:
\lim_{x \to -\infty} \frac{|x|}{e^{-x}}

As x tends to negative infinity, -x tends to infinity.
If x = -1000, then -x = 1000.
As x approaches negative infinity, -x approaches infinity, and \frac{|x|}{e^{-x}} approaches 0.

Shorthand Notation for Infinite Limits

Although the limit does not technically exist, the notation \lim_{x \to \infty} f(x) = \infty is used to indicate that the function grows unboundedly.

Examples:

Case 1:

*\lim_{x \to \infty} f(x) = \infty

Example: y = \sqrt{x}

Case 2:

*\lim_{x \to \infty} f(x) = -\infty

Example: y = -\sqrt{x}

Case 3:

*\lim_{x \to -\infty} f(x) = \infty

Example: y = e^{-x}

Case 4:

*\lim_{x \to -\infty} f(x) = -\infty

Example: y = -e^{-x}
In Angsten 90, use the notation "does not exist" when the limit is infinite, but describe the behavior of the function (e.g., grows unboundedly).
Arithmetic with infinity (e.g., \infty - \infty, \frac{\infty}{\infty}) is not well-defined.
However, \frac{1}{\infty} = 0 is an intuitive concept.

Asymptotes

Vertical Asymptotes

The line x = a is a vertical asymptote for the curve y = f(x) if at least one of the following is true:
*\lim{x \to a} f(x) = \infty *\lim{x \to a} f(x) = -\infty
*\lim{x \to a^-} f(x) = \infty *\lim{x \to a^-} f(x) = -\infty
*\lim{x \to a^+} f(x) = \infty *\lim{x \to a^+} f(x) = -\infty

*Example:
f(x) = \frac{1}{x}

\lim_{x \to 0^-} f(x) = -\infty
\lim_{x \to 0^+} f(x) = \infty
The function has a vertical asymptote at x = 0.

Horizontal Asymptotes

The line y = L is a horizontal asymptote for the curve y = f(x) if:
*\lim{x \to \infty} f(x) = L *\lim{x \to -\infty} f(x) = L

*Example:
f(x) = \frac{x^2 - 1}{x^2 + 1}

\lim{x \to \infty} f(x) = 1 \lim{x \to -\infty} f(x) = 1
So, we can see that y = 1 is a horizontal asymptote.
The graph of the function approaches 1 as x approaches infinity or negative infinity.
A function can cross a horizontal asymptote at some points. The horizontal asymptote describes the behavior of the function at very large values of x.