Problem 81
Question
Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+25}-\sqrt{x^{2}-1}\right)$$
Step-by-Step Solution
Verified Answer
The limit is 0.
1Step 1: Define the Limit Expression
We start by writing the limit expression that we need to solve: \( \lim_{x \to \infty}\left(\sqrt{x^2 + 25} - \sqrt{x^2 - 1}\right) \).
2Step 2: Simplify the Expression Using Conjugates
To simplify, multiply and divide by the conjugate: \( \lim_{x \to \infty} \frac{\left(\sqrt{x^2 + 25} - \sqrt{x^2 - 1}\right)\left(\sqrt{x^2 + 25} + \sqrt{x^2 - 1}\right)}{\sqrt{x^2 + 25} + \sqrt{x^2 - 1}} \).\The numerator becomes \((x^2 + 25) - (x^2 - 1) = 26\).
3Step 3: Simplify the Limit Expression
We then simplify the expression:\[\lim_{x \to \infty} \frac{26}{\sqrt{x^2 + 25} + \sqrt{x^2 - 1}}\]Substituting for infinity, this gives us \( \frac{26}{2x} \).
4Step 4: Evaluate the Limit
As \(x\) approaches infinity, \( \frac{26}{2x} \to 0 \). This simplifies as \( x \to \infty \).
5Step 5: Conclusion
Therefore, the limit of the expression is 0.
Key Concepts
Infinite LimitsConjugate MethodLimit Evaluation
Infinite Limits
In calculus, the term infinite limits refers to limits that explore the behavior of a function as the variable approaches infinity, either positive or negative, or when the function itself grows without bound. As functions can behave differently at infinity, understanding infinite limits is crucial in analyzing such behavior.
For the problem at hand, we are interested in the behavior of the expression \( \lim_{x \to \infty}(\sqrt{x^2 + 25} - \sqrt{x^2 - 1}) \) as \( x \) approaches positive infinity. Essentially, we want to determine what happens to the target expression when \( x \) becomes very large. The expression involves two square roots that individually grow as \( x \) becomes larger. However, when subtracted, more subtle behavior emerges, requiring special techniques to calculate. In the real world, such questions about infinite behavior can often arise in physics, engineering, and even economics where processes continue indefinitely.
Understanding this aspect of limits helps in grasping how certain mathematical functions stabilize or maintain a consistent trajectory over time.
For the problem at hand, we are interested in the behavior of the expression \( \lim_{x \to \infty}(\sqrt{x^2 + 25} - \sqrt{x^2 - 1}) \) as \( x \) approaches positive infinity. Essentially, we want to determine what happens to the target expression when \( x \) becomes very large. The expression involves two square roots that individually grow as \( x \) becomes larger. However, when subtracted, more subtle behavior emerges, requiring special techniques to calculate. In the real world, such questions about infinite behavior can often arise in physics, engineering, and even economics where processes continue indefinitely.
Understanding this aspect of limits helps in grasping how certain mathematical functions stabilize or maintain a consistent trajectory over time.
Conjugate Method
The conjugate method is an algebraic technique used primarily to simplify expressions involving roots or fractions. It involves multiplying the original expression by a conjugate, which is a binomial formed by changing the sign between two terms in a radical expression.
Applying this method helps us tackle the original expression \( \sqrt{x^2 + 25} - \sqrt{x^2 - 1} \) by pairing it with its conjugate: \( \sqrt{x^2 + 25} + \sqrt{x^2 - 1} \). This process yields a rational expression by eliminating square roots and simplifying the numerator dramatically. Here is how it looks step by step:
Applying this method helps us tackle the original expression \( \sqrt{x^2 + 25} - \sqrt{x^2 - 1} \) by pairing it with its conjugate: \( \sqrt{x^2 + 25} + \sqrt{x^2 - 1} \). This process yields a rational expression by eliminating square roots and simplifying the numerator dramatically. Here is how it looks step by step:
- Multiply and divide the original expression by its conjugate.
- This results in \( \frac{(\sqrt{x^2 + 25} - \sqrt{x^2 - 1})(\sqrt{x^2 + 25} + \sqrt{x^2 - 1})}{\sqrt{x^2 + 25} + \sqrt{x^2 - 1}} \).
- Simplify to get \( \frac{26}{\sqrt{x^2 + 25} + \sqrt{x^2 - 1}} \).
Limit Evaluation
Evaluating limits involves determining the value that a function approaches as the input approaches a specified point. In this case, we achieved simplification by using the conjugate to rewrite the expression in a more workable form.
After reducing the problem to \( \lim_{x \to \infty} \frac{26}{\sqrt{x^2 + 25} + \sqrt{x^2 - 1}} \), we analyze the behavior as \( x \to \infty \). In this simplified expression, the denominator approximately equals \( 2x \), since the terms inside the square roots become dominated by \( x^2 \).
After reducing the problem to \( \lim_{x \to \infty} \frac{26}{\sqrt{x^2 + 25} + \sqrt{x^2 - 1}} \), we analyze the behavior as \( x \to \infty \). In this simplified expression, the denominator approximately equals \( 2x \), since the terms inside the square roots become dominated by \( x^2 \).
- Substitute in the expression for large values of \( x \).
- The expression becomes \( \frac{26}{2x} \), which further simplifies with \( x \) in the denominator.
- As \( x \rightarrow \infty \), \( \frac{26}{2x} \to 0 \), implying the original limit evaluates to 0.
Other exercises in this chapter
Problem 80
Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow \infty}(\sqrt{x+9}-\sqrt{x+4})$$
View solution Problem 81
a. Graph \(g(x)=x \sin (1 / x)\) to estimate lim \(_{x \rightarrow 0} g(x),\) zooming in on the origin as necessary. b. Confirm your estimate in part (a) with a
View solution Problem 82
a. Graph \(h(x)=x^{2} \cos \left(1 / x^{3}\right)\) to estimate \(\lim _{x \rightarrow 0} h(x),\) zooming in on the origin as necessary. b. Confirm your estimat
View solution Problem 82
Find the limits in Exercises \(80-86\). $$\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}+3}+x\right)$$
View solution