Problem 28 - Calculus: Early Transcendentals

Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and \(k\) are fixed real numbers. $$\lim _{x \rightarrow 2}\left(x^{2}-x\right)^{5}$$

Step-by-Step Solution

Answer

Answer: The limit as x approaches 2 of the function $(x^2 - x)^5$ is 32.

Step 1: Simplify the expression

First, simplify the expression inside the parentheses: $$(x^{2}-x)^{5} = [(x(x-1))]^{5}$$

Step 2: Evaluate the limit

Now we can find the limit as x approaches 2: $$\lim _{x \rightarrow 2}\left(x^{2}-x\right)^{5} = \lim _{x \rightarrow 2}\left[(x(x-1))^5\right]$$ Because polynomial functions are continuous, we can simply plug in x=2 directly to find the limit: $$\lim _{x \rightarrow 2}\left[(x(x-1))^5\right] = \left[(2(2-1))^5\right]$$ $$ = (2)^5$$ $$ = 32$$ Thus, the limit as x approaches 2 of the given expression is 32.

Key Concepts

Continuity of Polynomial Functions

One of the fundamental properties of polynomial functions is that they are continuous. Continuity, in general, refers to the idea that a function doesn't have any breaks, jumps, or holes in its graph when plotted over its domain. This smoothness allows us to evaluate limits fairly straightforwardly.

Polynomials are made up of terms that include a constant, an unknown variable raised to a whole number exponent, and possibly multiplied by a coefficient. The nature of these terms and the fact that they are summed together ensures that when you plot their graphs, the curve does not suddenly start or stop or have any abrupt changes in direction.

Visualizing Continuity

In the context of continuity, if you were to draw the graph of a polynomial function, you could do so without picking up your pencil. This unbroken flow is what makes working with polynomials and limits much simpler compared to functions that are not continuous, which often require more complex analysis to evaluate their limits.

Evaluating Limits

Evaluating limits is a central operation in calculus, used to determine the value that a function approaches as the input (usually denoted as 'x') gets closer to a specific point. The notation \( \lim_{{x \rightarrow a}} f(x) \) represents the limit of the function \( f(x) \) as \( x \) approaches the value \( a \).

When dealing with polynomial functions and their limits, the procedure is greatly simplified thanks to their continuity. To evaluate the limit of a polynomial as \( x \) approaches some value \( a \), you can often substitute \( a \) directly into the polynomial in place of \( x \) and simplify. This step is commonly referred to as 'plugging in' the value.

Direct Substitution

Direct substitution is possible with continuous functions, like polynomials, because there's no concern about a break or hole at the point \( x = a \). By just replacing \( x \) with \( a \) in the expression and simplifying, you arrive at the limit. This method stands in contrast to more complex functions, where additional steps may be necessary to find the limit.

Limits of Polynomials

In the context of limits, polynomials exhibit very user-friendly behavior. The limit of a polynomial function as \( x \) approaches any given number can be found simply because the polynomial function is continuous over its entire domain, which includes all real numbers.

Take the exercise example, where we evaluated the limit of \( (x^2 - x)^5 \) as \( x \) approaches 2. Due to the continuous nature of polynomials, we were able to plug in the value 2 directly into the expression and simplify to find the limit, which resulted in 32. This procedure underscores an important concept: the limit of a polynomial function at a given point is equal to the value of the function at that point.

Power of Simplicity

One of the strengths of polynomial functions is their predictability and the ease with which we can find limits. This simplicity becomes especially valuable when polynomials are used to model real-world situations where knowing the behavior of a function at a specific point or as it approaches that point is crucial. The straightforward evaluation of their limits makes polynomials a powerful tool in various applications across mathematics, physics, engineering, and economics.