Problem 10 - Calculus Early Transcendentals

Question

Evaluate the following limits. $$\lim _{x \rightarrow \infty}\left(5+\frac{1}{x}+\frac{10}{x^{2}}\right)$$

Step-by-Step Solution

Answer

Question: Evaluate the limit of the expression below as x approaches infinity: $$\lim_{x \rightarrow \infty} \left(5+\frac{1}{x}+\frac{10}{x^{2}}\right)$$ Answer: 0

Step 1: Identify the highest power of x

In the given expression, the highest power of x is \(x^2\), which is found in the term \(\frac{10}{x^2}\).

Step 2: Divide each term by the highest power of x

We will divide each term of the expression by \(x^2\): $$\frac{5}{x^2} + \frac{1}{x^3} + \frac{10}{x^4}$$

Step 3: Evaluate the limit as x approaches infinity

Now, we evaluate the limit of each term as x approaches infinity: $$\lim_{x \rightarrow \infty} \frac{5}{x^2} = 0$$ $$\lim_{x \rightarrow \infty} \frac{1}{x^3} = 0$$ $$\lim_{x \rightarrow \infty} \frac{10}{x^4} = 0$$

Step 4: Combine the limits of each term

Since the limit of each term as x approaches infinity is 0, the overall limit of the expression is also 0: $$\lim_{x \rightarrow \infty} \left(5+\frac{1}{x}+\frac{10}{x^{2}}\right) = 0$$

Key Concepts

Infinity

In calculus, infinity is a concept that describes something without any bound or limit. When we talk about infinity in terms of limits, we're considering what happens as a variable grows larger and larger.
For example, when we say "as x approaches infinity," we're examining the behavior of a function as x becomes indefinitely large. It's interesting to note that infinity is not a number, but a way to express an unbounded value. This means that operations involving infinity, such as division or addition, follow their own rules different from those of real numbers.
Using limits, we can explore how functions behave as they move towards infinite values. Particular rules and techniques, like dividing every term by the highest power of the variable, help simplify and calculate these expressions.

Rational Functions

Rational functions are a fundamental concept in algebra, characterized by a quotient of two polynomials.
These functions can be written in the form \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials. The behavior of rational functions is particularly interesting because it involves the concept of limits and asymptotes, which emerges when evaluating them near certain values or at infinity.
The degree of the polynomials involved plays a crucial role in determining the behavior of the rational function. For instance:

• If the degree of \( P(x) \) is less than \( Q(x) \), as \( x \) approaches infinity, the function approaches zero.
• If the degrees are equal, the function approaches a finite value dictated by the leading coefficients of \( P(x) \) and \( Q(x) \).
• If the degree of \( P(x) \) is greater, the function may approach infinity or negative infinity, depending on the sign of the leading coefficients and degree differences.

Understanding these behaviors can help predict how rational functions will act, which is essential for solving limits and understanding asymptotic behavior.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches a particular value, often infinity.
In rational functions, this term refers to the tendency of a function to get closer to a line (an asymptote) as the input gets to extremely high or low values. For vertical asymptotes, the function tends to infinity, whereas for horizontal asymptotes, the function approaches a constant value.
When evaluating limits, we often look for asymptotic behavior to simplify the understanding of complex functions. This method checks if parts of the function vanish or dominate as they approach certain extreme values.

• Horizontal asymptotes occur when the output of the function approaches a constant value as \( x \) approaches infinity or negative infinity.
• Vertical asymptotes denote places where the function goes towards infinity as \( x \) approaches a particular finite value.
• Oblique asymptotes may arise when a polynomial in a rational expression has a degree higher than the polynomial in the denominator by exactly one degree.

Recognizing these behaviors allows you to predict how functions behave in extreme conditions, simplifying complex limit evaluations.