Problem 31 - Calculus: Early Transcendentals

Question

Evaluate limit and justify your answer. $$\lim _{x \rightarrow 0}\left(x^{8}-3 x^{6}-1\right)^{40}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit of the given function as x approaches 0: $$\lim_{x \rightarrow 0} \left( x^8 - 3x^6 - 1 \right)^{40}$$ Answer: The limit of the given function as x approaches 0 is 1.

Step 1: Focus on the function f(x)

We first consider the function inside the power: $$f(x) = x^8 - 3x^6 - 1$$ Taking the limit of this function as \(x \rightarrow 0\), we have: $$\lim_{x \rightarrow 0} f(x) = \lim_{x \rightarrow 0} (x^8 - 3x^6 - 1)$$

Step 2: Evaluate the limit of f(x)

Evaluating the limit as \(x\) approaches 0: $$\lim_{x \rightarrow 0} (x^8 - 3x^6 - 1) = (0)^8 - 3(0)^6 - 1 = 0 - 0 - 1 = -1$$ So, when \(x \rightarrow 0\), the function \(f(x)\) approaches \(-1\).

Step 3: Evaluate the limit of \((f(x))^n\)

Now that we know the limit of the function \(f(x)\) as \(x\) approaches 0 is \(-1\), we can find the limit of the given expression, i.e., \((f(x))^n\) with \(n=40\): $$\lim_{x \rightarrow 0} \left( x^8 - 3x^6 - 1 \right)^{40} = \lim_{x \rightarrow 0} (-1)^{40}$$

Step 4: Justify the answer

Since \((-1)^{40}\) is a constant and does not depend on \(x\), the limit will be equal to that constant itself: $$\lim_{x \rightarrow 0} (-1)^{40} = (-1)^{40} = 1$$ The limit, as \(x\) approaches 0, of the given function \(\left(x^{8}-3 x^{6}-1\right)^{40}\) is 1.

Key Concepts

Power Functions

Power functions are expressions where a variable is raised to an exponent. They form the foundation for many mathematical concepts, including polynomial functions. In our problem, the power function is emphasized in the expression \(x^8 - 3x^6 - 1\), raised to the power of 40. This means we are not only dealing with a polynomial but are deeply exploring its behavior as the input variable \(x\) changes.

When working with power functions, the role of the exponent is crucial. Large exponents, like 40 in this exercise, significantly influence the behavior of the function, especially near zero or for small changes in \(x\). For a power function that is negative inside but raised to an even exponent, like -1 to the power of 40, the result becomes positive. Understanding these nuances helps decode complex mathematical limits and analyze their behavior across different scenarios.

• Power functions can expand our understanding of various numerical phenomena.
• They are crucial for analyzing limits and continuity.
• High exponents can drastically alter the value of outcomes.

Algebraic Limits

Algebraic limits deal with evaluating the behavior of algebraic expressions as the variable approaches a certain value. In our exercise, we looked at the limit as \(x\) approaches 0 for the expression \(x^8 - 3x^6 - 1\). These types of limits often require simplifying the expression or directly substituting the approaching value into the function.

For algebraic limits, substitution is a handy technique if the function is continuous at the given point. As seen in our solution, substituting \(x = 0\) into the polynomial gives \(-1\). Algebraic manipulation, such as simplifying the expression before evaluating the limit, is vital to understand how each term contributes to the overall limit. This forms a basis for solving more complex calculus problems.

• Algebraic limits involve evaluating polynomial forms.
• Simplification is often necessary before limit evaluation.
• Substitution works well for continuous functions at a point.

Constant Function Limit

A constant function limit refers to a situation where the limit of a function results in a constant. After determining \(f(x)\) approaches -1, the outcome of the power function \(f(x)^{40}\) results in a constant: 1. This simplification occurs due to the power rule used effectively after knowing the expression simplifies to a constant.

Constant function limits are straightforward yet essential for understanding larger mathematical concepts, particularly in calculus. They simplify analysis, as the focus shifts from variable-dependent expressions to fixed numeric outcomes. In this exercise, since \((-1)^{40} = 1\), the limit is a result of appreciating this property of powers and constant evaluation.

• Constant limits result in fixed outputs regardless of the input variable.
• They often come from simplifying polynomial expressions.
• They help comprehend broader mathematical principles efficiently.