Problem 30 - Calculus Early Transcendentals

Question

15-36 Find the limit. $$ \lim _{x \rightarrow \infty} \sqrt{x^{2}+1} $$

Step-by-Step Solution

Answer

The limit is infinity.

Step 1: Analyze the Expression

We need to find the limit of \( \sqrt{x^2 + 1} \) as \( x \) approaches infinity. The expression contains a square root, which might complicate direct substitution.

Step 2: Simplify the Expression

Divide the terms inside the square root by \( x^2 \) to simplify the expression:\[ \sqrt{x^2 + 1} = \sqrt{x^2(1 + \frac{1}{x^2})} \]This can further be broken down using the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \) to:\[ \sqrt{x^2} \cdot \sqrt{1 + \frac{1}{x^2}} = x \cdot \sqrt{1 + \frac{1}{x^2}} \] assuming \( x \) approaches infinity and is positive.

Step 3: Evaluate the Limit

Now, we analyze the new expression. As \( x \rightarrow \infty \), the term \( \frac{1}{x^2} \rightarrow 0 \). Thus, the expression simplifies to:\[ x \cdot \sqrt{1 + \frac{1}{x^2}} \approx x \cdot \sqrt{1} = x \]

Step 4: Conclude with the Limit Result

Since \( x \) approaches infinity, \( x \) itself will approach infinity. Thus, the limit of \( \sqrt{x^2+1} \) as \( x \rightarrow \infty \) is infinity.

Key Concepts

Infinite Limits

Infinite limits occur when the value of a function becomes larger and larger, without bound, as the input grows. Understanding infinite limits is crucial when analyzing the behavior of functions as they approach extreme values, like infinity. When dealing with expressions involving infinitely large values, such as \( x \rightarrow \infty \), direct substitution does not work as it might in simpler limit problems.To handle these cases, we often simplify the expression to see how different parts of the equation behave as \( x \) increases. This strategy can help determine the overall behavior of the function. For example, in the expression \( \sqrt{x^2+1} \), we see that as \( x \) approaches infinity, the internal expression \( x^2+1 \) becomes large. Through simplification, we can better understand how to approach infinite limits.

Square Root Simplification

Simplifying expressions involving square roots can be tricky, but it's an essential algebraic skill often used in calculus. The square root function itself grows at a slower rate compared to polynomial functions, which can change how limits are calculated.In the given problem, we have \( \sqrt{x^2 + 1} \). Direct evaluation when \( x \) tends to infinity is not straightforward.However, it can be simplified by factoring out \( x^2 \) from under the square root:

• First, rewrite the expression as \( \sqrt{x^2(1 + \frac{1}{x^2})} \).
• Use the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \) to separate the terms into \( \sqrt{x^2} \cdot \sqrt{1 + \frac{1}{x^2}} \).
• Since \( \sqrt{x^2} = x \) when \( x \) is positive, the expression simplifies to \( x \cdot \sqrt{1 + \frac{1}{x^2}} \).

This process helps transform the original expression into a more manageable form for evaluating limits.

Asymptotic Behavior

Asymptotic behavior refers to how functions behave as they move towards an extreme value or infinity.Understanding this concept can provide valuable insight into the long-term trend or approach of a function.In our example, after simplifying the expression to \( x \cdot \sqrt{1 + \frac{1}{x^2}} \), we can observe the asymptotic behavior as \( x \) increases. When considering the term \( \sqrt{1 + \frac{1}{x^2}} \):

• As \( x \rightarrow \infty \), \( \frac{1}{x^2} \rightarrow 0 \).
• Thus, the term \( \sqrt{1 + \frac{1}{x^2}} \rightarrow \sqrt{1} = 1 \).

Combined with the \( x \), you see that the leading behavior of \( x \cdot \sqrt{1 + \frac{1}{x^2}} \) ultimately resembles \( x \), an asymptote towards infinity.This knowledge helps predict how changes in \( x \) influence the function long-term, linking to the broader calculus concept of asymptotic relations.