Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \lim _{x \rightarrow 0} x^{4}=0 $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Limit Expression

The limit expression \( \lim_{x \rightarrow 0} x^4 \) asks us to find the value that \( x^4 \) approaches as \( x \) gets closer and closer to zero.

Step 2: Identify the Function

The expression \( x^4 \) is a polynomial function. A polynomial function raised to any power will be continuous and defined at all real number inputs.

Step 3: Substitute the Limit Point

Since \( x^4 \) is continuous at \( x=0 \), you can substitute \( x=0 \) directly into the function: \( (0)^4 = 0 \).

Step 4: Conclusion from Substitution

After substituting \( x=0 \) into the expression \( x^4 \), we find the resultant value is 0. Therefore, as \( x \) approaches 0, \( x^4 \) also approaches 0.

Key Concepts

Polynomial Functions

A polynomial function is a type of mathematical expression made up of variables and coefficients. It involves only the operations of addition, subtraction, multiplication, and non-negative integer exponentiation of variables. These functions are expressed in a standard form like this:

• \( a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 \, \)

where \( a_n, a_{n-1}, \dots, a_0 \) are coefficients and \( n \) is a non-negative integer.
One key characteristic of polynomial functions is their continuity. This means unlike functions with sharp turns or breaks, polynomial functions are smooth and unbroken.
In the original exercise where we have \( x^4 \, \) it's a polynomial function of degree 4, which ensures a smooth graph without any interruptions. This property plays well when calculating limits, as polynomials tend to behave predictably compared to other types of functions.

Continuity

Continuity is an important concept in calculus that describes the behavior of functions. A function is considered continuous at a point when there are no breaks, jumps, or holes in its graph at that particular point.
More formally, a function \( f(x) \) is continuous at a value \( x = c \) if the following conditions are satisfied:

• \( f(c) \) is defined.
• The limit \( \lim_{x \to c} f(x) \) exists.
• \( \lim_{x \to c} f(x) = f(c) \)

Continuity ensures that small changes in input bring about small changes in output, implying the graph can be drawn without lifting a pencil.
For polynomial functions like \( x^4 \, \) continuity is guaranteed everywhere on the real number line. This is because polynomial functions do not have any breaks or undefined points, making them easy to handle when calculating limits.

Direct Substitution Method

The direct substitution method is a basic but effective approach for finding limits, particularly useful for continuous functions. This method works on the principle that if a function \( f(x) \) is continuous at a point \( x = c \), then the limit of the function as \( x \) approaches \( c \) is simply the value of the function at \( c \).
Mathematically, this is expressed as:

• \( \lim_{x \to c} f(x) = f(c) \) when \( f(x) \) is continuous at \( c \)

In our example, for \( \lim_{x \to 0} x^4 \, \) since \( x^4 \) is a polynomial function and therefore continuous everywhere, we simply substitute \( x = 0 \) into it to find \( (0)^4 = 0 \).
Thus, the direct substitution method offers a straightforward route to calculating limits whenever the function involved is known to be continuous at the point of interest.