Problem 10 - Contemporary Precalculus

Question

Find the limit if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow-1}\left(x^{7}+2 x^{5}-x^{4}+3 x+4\right)$$

Step-by-Step Solution

Answer

Answer: The limit of the function as x approaches -1 is 0.

Step 1: Replace x with -1 in the function

Start by plugging in -1 wherever we see x in the given function: $$\lim _{x \rightarrow -1}\left((-1)^{7}+2 (-1)^{5}-(-1)^{4}+3 (-1)+4\right)$$

Step 2: Simplify the terms

Now, calculate each term with x replaced by -1: $$=(-1)^7 + 2(-1)^5 - (-1)^4 + 3(-1) + 4$$ $$=(-1) + 2(-1) - 1 - 3 + 4$$

Step 3: Calculate the limit

With the terms simplified, compute the numerical value of the limit: $$=(-1) + (-2) -1 - 3 + 4$$ $$=-1-2-1-3+4 = -1+1 = 0$$ The limit of the function as x approaches -1 is 0: $$\lim _{x \rightarrow-1}\left(x^{7}+2 x^{5}-x^{4}+3 x+4\right)=0$$

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables, coefficients, and non-negative integer exponents. They are written in the general form:
\[ a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \] where:

• \( a_n, a_{n-1}, \ldots, a_1, a_0 \) are coefficients and can be any real number.
• \( n \) is a non-negative integer that represents the highest power of the variable \( x \), and it's called the degree of the polynomial.

A polynomial function can have one or multiple terms, and these terms are simply the individual \( x^i \) components with their coefficients. For example, in our exercise, \( x^7 \) and \( 3x \) are individual terms of the function. Polynomials are widely used due to their simple structure and the ability to model various real-world phenomena. They play a significant role in calculus, especially in examining the behavior of functions as the input values change.

Evaluating Limits

Limits are a fundamental concept in calculus, used to understand the behavior of functions as they approach a particular point. To find a limit, specifically an expression like \( \lim_{x \to c} f(x) \), we need to determine the value that \( f(x) \) approaches as \( x \) gets infinitely close to \( c \).
When evaluating limits, there are a few basic approaches:

• Direct Substitution: If the function is continuous at \( x = c \), the simplest method is direct substitution. This means you replace \( x \) with \( c \) in the equation, like in our example with \( x = -1 \).
• Factoring and Simplifying: If direct substitution leads to an indeterminate form like \( \frac{0}{0} \), sometimes factoring the function can help simplify it into a form that allows evaluation.
• Graphical or Numerical Approaches: For more complex functions, graphing or numerical tables can provide insight into the behavior as \( x \) approaches the value.

In our exercise, since we are dealing with a polynomial function, direct substitution was sufficient to determine the limit.

Limit Laws

Limit laws are a set of rules that allow us to evaluate limits for more complex functions using simpler operations. These laws apply when the limits of individual functions within an expression are known.
Here are some useful limit laws:

• Sum Law: If \( \lim_{x \to c} f(x) = L \) and \( \lim_{x \to c} g(x) = M \), then \( \lim_{x \to c} [f(x) + g(x)] = L + M \).
• Difference Law: This is similar to the sum law. If \( \lim_{x \to c} f(x) = L \) and \( \lim_{x \to c} g(x) = M \), it follows that \( \lim_{x \to c} [f(x) - g(x)] = L - M \).
• Product Law: If \( \lim_{x \to c} f(x) = L \) and \( \lim_{x \to c} g(x) = M \), then \( \lim_{x \to c} [f(x)g(x)] = L \times M \).
• Quotient Law: If \( \lim_{x \to c} f(x) = L \) and \( \lim_{x \to c} g(x) = M \) and \( M eq 0 \), then \( \lim_{x \to c} \left(\frac{f(x)}{g(x)}\right) = \frac{L}{M} \).
• Power Law: If \( \lim_{x \to c} f(x) = L \) and \( n \) is a positive integer, then \( \lim_{x \to c} [f(x)]^n = L^n \).

These laws are incredibly helpful for breaking down more complicated limits into simpler parts that can be easily managed. They were not needed extensively in our exercise, due to the straightforward nature of polynomial functions, but knowing them can simplify more complex problems involving limits.