1.14 Connecting Infinite Limits and Vertical Asymptotes

In this topic, we will focus on understanding the behavior of functions as they approach infinity. We will explore the concept of infinite limits and how they relate to vertical asymptotes in a function.

Objectives:

Essential Knowledge:

When a function approaches a value of positive or negative infinity, the limit is said to be infinite.

For example, if a function f(x) approaches positive infinity as x approaches a certain value, we would write the limit as:

lim x->a f(x) = +infinity

Similarly, if a function approaches negative infinity, we would write the limit as:

lim x->a f(x) = -infinity

In both cases, the limit does not exist in the traditional sense, as the function does not approach a specific value. Instead, the limit tells us about the behavior of the function as x approaches a certain value.

Vertical asymptotes are another way to describe the behavior of a function as it approaches infinity. A vertical asymptote occurs when the function becomes infinitely large (positive infinity) or infinitely small (negative infinity) at a certain value of x.

For example, a function with a vertical asymptote at x = a would be written as:

f(x) = 1/(x-a)

As x approaches the value of a, the denominator (x-a) becomes smaller and smaller, causing the value of the function to become larger and larger. This results in a vertical asymptote at x = a.

Let's look at some examples to further illustrate the concept of infinite limits and vertical asymptotes:

1. f(x) = 1/x, as x approaches 0, the function approaches positive infinity, and we would write the limit as: lim x->0 f(x) = +infinity As x moves closer to 0, the denominator becomes smaller and smaller, causing the function to become infinitely large. This results in a vertical asymptote at x = 0.

2. g(x) = 1/x^2, as x approaches infinity, the function approaches 0, and we would write the limit as: lim x->infinity g(x) = 0 This function has a horizontal asymptote at y = 0, as the value of the function approaches 0 as x approaches infinity.

3. h(x) = (x-1)/(x^2+1), as x approaches 1, the function approaches 0, and we would write the limit as: lim x->1 h(x) = 0 This function has a hole at x = 1, as the function approaches 0, but is not defined at that point.

4. j(x) = x^2-2x-3/x-3, as x approaches 3, the function approaches -1, and we would write the limit as: lim x->3 j(x) = -1 As x moves closer to 3, the denominator gets closer to 3, causing the function to approach -1. This function has a vertical asymptote at x = 3.

5. k(x) = 1/x+2, as x approaches -2, the function approaches negative infinity, and we would write the limit as: lim x->-2 k(x) = -infinity As x moves closer to -2, the denominator gets closer to -2, causing the function to approach negative infinity. This results in a vertical asymptote at x = -2.

6. l(x) = (x^2-1)/(x-1), as x approaches 1, the function approaches positive infinity, and we would write the limit as: lim x->1 l(x) = +infinity As x moves closer to 1, the denominator gets closer to 0, causing the function to approach positive infinity. This results in a vertical asymptote at x = 1.

7. m(x) = (x-2)/(x^2+4), as x approaches 2, the function approaches 0, and we would write the limit as: lim x->2 m(x) = 0 This function has a hole at x = 2, as the function approaches 0, but is not defined at that point.

8. n(x) = (x^2-4)/(x-2), as x approaches 2, the function approaches positive infinity, and we would write the limit as: lim x->2 n(x) = +infinity As x moves closer to 2, the denominator gets closer to 0, causing the function to approach positive infinity. This results in a vertical asymptote at x = 2.

9. p(x) = x^3-9x+3/x-3, as x approaches 3, the function approaches 4, and we would write the limit as: lim x->3 p(x) = 4 This function has a vertical asymptote at x = 3.

10. q(x) = (x^2-4)/(x^2-16), as x approaches 2 and -4, the function approaches 1 and -1 respectively, and we would write the limit as: lim x->2 q(x) = 1, lim x->-4 q(x) = -1 As x moves closer to 2 and -4, the denominator gets closer to 0, causing the function to approach 1 and -1 respectively. This results in vertical asymptotes at x = 2 and x = -4.

In summary, understanding infinite limits and vertical asymptotes is crucial for interpreting the behavior of functions as they approach infinity. By using limits and analyzing the behavior of functions, we can gain a better understanding of the asymptotic and unbounded behavior of functions. It is important to note that when a function has a vertical asymptote it does not exists at that point, and the limit is either positive or negative infinity.