Understanding Limits at Infinity and Limits of Indeterminate Forms in Calculus: A Comprehensive Guide

Two Special Limits

There are two special types of limits in calculus that are commonly encountered: limits at infinity and limits of indeterminate forms

There are two special types of limits in calculus that are commonly encountered: limits at infinity and limits of indeterminate forms.

1. Limits at Infinity:
Limits at infinity can be categorized into three possible scenarios, depending on the behavior of the function as the input approaches infinity:

a) If the function approaches a specific value as the input approaches infinity, then the limit exists and is equal to that specific value. Mathematically, this can be expressed as:

lim_ (x -> ∞) f(x) = L

For example, consider the function f(x) = 1/x. As x increases without bound, f(x) approaches 0. Hence, the limit of f(x) as x approaches infinity is:

lim_ (x -> ∞) 1/x = 0

b) If the function grows without bound as the input approaches infinity, then the limit is said to be divergent (or does not exist). Mathematically, this can be expressed as:

lim_ (x -> ∞) f(x) = ∞

For example, consider the function f(x) = x^2. As x becomes arbitrarily large, f(x) also becomes arbitrarily large. Therefore, the limit of f(x) as x approaches infinity is:

lim_ (x -> ∞) x^2 = ∞

c) If the function oscillates between different values as the input approaches infinity, then the limit is also said to be divergent (or does not exist). In this case, the limit may not be a specific value or infinity. Instead, it can be expressed as:

lim_ (x -> ∞) f(x) = DNE

For example, consider the function f(x) = sin(x). As x increases without bound, sin(x) oscillates between -1 and 1. Since the function does not approach a specific value or infinity, the limit of f(x) as x approaches infinity is:

lim_ (x -> ∞) sin(x) = DNE (Does Not Exist)

2. Limits of Indeterminate Forms:
Limits of indeterminate forms occur when the function exhibits an algebraic ambiguity when evaluating the limit. These expressions typically involve the form 0/0, ∞/∞, or ∞ – ∞. In such cases, additional techniques like L’Hôpital’s Rule or algebraic manipulation may be used to evaluate the limit.

For instance, consider the following indeterminate form:

lim_ (x -> 1) (x^3 – 1)/(x – 1)

If we directly substitute x = 1 into the expression, we get 0/0, which is an indeterminate form. However, if we factorize the numerator and reduce the expression, we obtain:

lim_ (x -> 1) (x^3 – 1)/(x – 1) = lim_ (x -> 1) (x – 1)(x^2 + x + 1)/(x – 1)

Now, we can cancel out the common factor (x – 1) and evaluate the limit:

lim_ (x -> 1) (x^2 + x + 1) = 1^2 + 1 + 1 = 3

Therefore, the value of the limit is 3.

In summary, these special limits play an important role in calculus and being familiar with their definitions and techniques to evaluate them is essential for solving various math problems.

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