Problem 31 - Calculus Early Transcendentals

Question

$$ \lim _{x \rightarrow \infty} \sqrt{x^{2}+1} $$ $$ \lim _{x \rightarrow-\infty}\left(x^{4}+x^{5}\right) $$

Step-by-Step Solution

Answer

For \( \lim_{x \to \infty} \sqrt{x^2 + 1} \), the limit is \( \infty \). For \( \lim_{x \to -\infty} (x^4 + x^5) \), the limit is also \( -\infty \).

Step 1: Determine the Form of the Expression for \( \lim_{x \to \infty} \sqrt{x^2 + 1} \)

As \( x \to \infty \), both \( x^2 \) and 1 grow large, however, \( x^2 \) dominates the expression because it grows much faster than 1. We can factor \( x^2 \) out of the square root to simplify the expression.

Step 2: Simplify the Expression for \( \lim_{x \to \infty} \sqrt{x^2 + 1} \)

Factor \( x^2 \) out of the square root: \( \sqrt{x^2 + 1} = \sqrt{x^2 (1 + \frac{1}{x^2})} = \sqrt{x^2} \sqrt{1 + \frac{1}{x^2}} = x \sqrt{1 + \frac{1}{x^2}} \). As \( x \to \infty \), \( \frac{1}{x^2} \to 0 \), so \( \sqrt{1 + \frac{1}{x^2}} \to 1 \).

Step 3: Find the Limit of \( \lim_{x \to \infty} x \sqrt{1 + \frac{1}{x^2}} \)

Since \( \sqrt{1 + \frac{1}{x^2}} \to 1 \) as \( x \to \infty \), the expression \( x \cdot 1 = x \). Thus, the limit is dominated by the term \( x \), therefore \( \lim_{x \to \infty} \sqrt{x^2 + 1} = \infty \).

Step 4: Determine the Form of the Expression for \( \lim_{x \to -\infty} (x^4 + x^5) \)

As \( x \to -\infty \), both \( x^4 \) and \( x^5 \) grow large, but \( x^5 \) grows faster than \( x^4 \).

Key Concepts

Infinite Limits

Exploring infinite limits is all about understanding how functions behave as the variable approaches infinity or negative infinity.
In calculus, when we say that a limit is approaching infinity, we aren't just saying that the function doesn't stop growing. We're also interested in how fast it grows and whether it reaches a stable pattern.
A common infinite limit situation is when you evaluate expressions like \[ \lim _{x \rightarrow \infty} \sqrt{x^{2}+1} \] Here, as \( x \rightarrow \infty \), the expression inside the square root, \( x^2 + 1 \), also becomes very large. However, this marks a situation where we need to figure out the dominant term, which helps us predict the behavior of the entire expression.

Dominant Term Analysis

Dominant term analysis is a key technique in evaluating limits that approach infinity. It enables us to identify the most significant component in an expression that dictates its growth as the variable becomes infinitely large.
In the expression \( \sqrt{x^2 + 1} \), the term \( x^2 \) is dominant because it grows much faster compared to the constant 1 as \( x \to \infty \).
When assessing dominant terms, determine which part of the expression increases the fastest:

• Compare growth rates: exponents, coefficients, and base values.
• Factor the dominant term to simplify calculations.

For instance, in \( \lim_{x \to -\infty} (x^4 + x^5) \), the term \( x^5 \) dominates because it grows exponentially faster than \( x^4 \). This term ultimately dictates the limit's behavior.

Simplifying Expressions

Simplifying expressions is a crucial step in solving limit problems efficiently. This method reduces complex expressions into simpler forms that are easier to analyze. Let's use the limit \( \lim_{x \to \infty} \sqrt{x^2 + 1} \) as an example.
To simplify, look to factor dominant terms, allowing unnecessary components to become negligible as \( x \to \infty \). Here’s how you'd handle the simplification:
1. Recognize the dominant term: \( x^2 \).2. Factor out \( x^2 \) from the square root: \[ \sqrt{x^2 + 1} = \sqrt{x^2(1 + \frac{1}{x^2})} = x \sqrt{1 + \frac{1}{x^2}} \] 3. As \( x \to \infty \), \( \frac{1}{x^2} \to 0 \), thus simplifying the expression to \( x \cdot 1 = x \).The clearer the expression, the more straightforward finding the limit becomes.

Factoring in Limits

Factoring is a powerful algebraic tool in limits that simplifies complex expressions by breaking them into manageable parts.
When dealing with polynomial expressions or square roots, factoring allows us to isolate dominant terms and provides a clearer understanding of the limit's behavior.
Take for example the expression \( \lim_{x \to \infty} \sqrt{x^2 + 1} \). By factoring, we rewrite it as:\[ \sqrt{x^2 (1 + \frac{1}{x^2})} = x \sqrt{1 + \frac{1}{x^2}} \] This shows:

• The function’s growth is tied to \( x \), as \( x \cdot 1 \to x \).
• Non-dominant terms like \( \frac{1}{x^2} \) tend to zero, simplifying calculations.

For \( \lim_{x \to -\infty} (x^4 + x^5) \), factoring confirms \( x^5 \) as the key growth driver—helping us find the limit accurately.