Problem 16 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(15 - 18\) , explain why you cannot use substitution to determine the limit. Find the limit if it exists. $$\lim _ { x \rightarrow 0 } \frac { 1 } { x ^ { 2 } }$$

Step-by-Step Solution

Answer

The limit of the function \( \frac { 1 } { x ^ { 2 } } \) as x approaches 0 is +∞.

Step 1: Understand the problem

We are given a rational function, \( f(x) = \frac { 1 } { x ^ { 2 } } \), and we are asked to find the limit of this function as \( x \) approaches 0.

Step 2: Determine if direct substitution will work

If we try to directly substitute \( x = 0 \) into the function, we will get \( \frac { 1 } { 0 } \), an undefined expression. Therefore, direct substitution is not an option here.

Step 3: Apply limit concepts

Notice that the function \( \frac { 1 } { x ^ { 2 } } \) approaches +∞ as x approaches 0 from the right ( \( x \rightarrow 0^+ \) ) and also +∞ as x approaches 0 from the left ( \( x \rightarrow 0^- \) ). Hence, we can say that the function approaches +∞ as x approaches 0.

Key Concepts

Direct Substitution

Direct substitution is a fundamental method in calculus used to find the limit of a function. It involves plugging the value that the variable approaches into the function to determine the limit. This method works well when the function is continuous at the point of interest, meaning there are no breaks, holes, or jumps in the graph of the function at that point.

However, with the exercise provided, direct substitution of 0 into the function, which gives us \( f(0) = \frac{1}{0^2} \) yields an undefined result. This is because division by zero is not permissible in mathematics; it breaks the fundamental rule that a number cannot be divided by zero. Therefore, direct substitution can't be used for functions that become undefined at the point of interest, especially when they lead to division by zero, as seen in the given rational function.

Rational Functions

A rational function is one that can be expressed as the ratio of two polynomials where the numerator and the denominator are polynomial functions. The key characteristic of these functions is that they are not defined when their denominator is equal to zero. In the task provided, \( f(x) = \frac{1}{x^2} \) is a rational function.

Rational functions are interesting because they can behave unpredictably near points where the denominator is zero. Here, as \( x \) approaches 0, the value of the function increases towards infinity, suggesting a vertical asymptote at \( x = 0 \) - a line that the graph of the function gets closer to but never actually touches. Understanding the behavior of rational functions near these critical points is important in calculus, as it helps to predict the behavior of the function around undefined points.

Approaching Infinity

The concept of limits involving infinity is a way to describe the behavior of functions as they grow without bound or decrease without end. When we say that a function approaches infinity, as \( x \) approaches a certain value, it means the function's value gets arbitrarily large.

In the case of the function \( \frac{1}{x^2} \) as \( x \) approaches 0, the value of the function grows significantly because the denominator of the fraction gets smaller and smaller, which makes the entire fraction value blow up. Mathematically, we describe this by saying that the limit of \( \frac{1}{x^2} \) as \( x \) approaches 0 is positive infinity, denoted as \( \lim _{x \rightarrow 0} \frac{1}{x^2} = +\infty \) whether \( x \) approaches from the left \( (x \rightarrow 0^-) \) or from the right \( (x \rightarrow 0^+) \) The understanding of a function's limit approaching infinity is crucial for comprehending its long-term behavior and ensuring proper mathematical analysis in calculus.