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Chapter 1: Problem 13

If \(\lim _{x \rightarrow \infty}(\sqrt{(x+a)(x+b)}-x)=f(a, b)\), then evaluate \(f(9,7)\)

Short Answer

Expert verified
The short answer is: \(f(9, 7) = 0\).

Step by step solution

01

Simplification of the expression inside the limit

The given expression inside the limit is: \[\sqrt{(x+a)(x+b)} - x\] We can try to simplify this expression by factoring out x. Firstly, let's rewrite the expression as: \[\sqrt{x^2 + (a+b)x + ab} - x\] Now, we can factor x out of both terms inside the square root: \[\sqrt{x^2(1 + \frac{a+b}{x} + \frac{ab}{x^2})} - x\]
02

Taking the limit as x approaches infinity

Now, we need to evaluate the limit as x approaches infinity: \[\lim_{x \rightarrow \infty} \left( \sqrt{x^2(1 + \frac{a+b}{x} + \frac{ab}{x^2})} - x\right)\] As x approaches infinity, the terms in the square root with x in the denominator approach zero: \[\lim_{x \rightarrow \infty} \frac{a+b}{x} = 0\] \[\lim_{x \rightarrow \infty} \frac{ab}{x^2} = 0\] With this in mind, the expression becomes: \[\lim_{x \rightarrow \infty} \left( \sqrt{x^2} - x\right) = \lim_{x \rightarrow \infty} (x - x) = 0\] Thus, the limit of the given expression as x approaches infinity is 0.
03

Evaluating f(a, b) at a = 9 and b = 7

Now that we have found the limit of the expression as x approaches infinity, we can evaluate f(a, b) at the given values a = 9 and b = 7. \[f(9, 7) = \lim_{x \rightarrow \infty} \left(\sqrt{(x+9)(x+7)} - x\right)\] As we found in step 2, this limit is equal to 0. Therefore, \[f(9, 7) = 0\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit of a Function
Understanding the limit of a function is fundamental in calculus. It describes the behavior of a function as the input approaches a certain value. For example, let鈥檚 consider the function \( f(x) \), its limit as \( x \) approaches \( c \) is the value that \( f(x) \) gets closer to as \( x \) gets infinitely close to \( c \). In the exercise, we are concerned with what happens as \( x \) becomes very large, which is a special kind of limit called an infinite limit. These concepts help us understand the long-term behavior of functions even when they don't actually reach a certain value.
Limits Approaching Infinity
In many calculus problems, we deal with functions as they grow very large, or as \( x \) approaches infinity. This type of limit helps us comprehend the end behavior of a function. The key takeaway is that things simplify when dealing with infinity. In our exercise, terms like \( (a+b)/x \) and \( ab/x^2 \) become negligible as \( x \) grows without bound. Therefore, these terms approach zero and can be eliminated from the expression when considering the limit as \( x \) approaches infinity. This eventually leads to an easier function to evaluate.
Simplification of Expressions

Algebra in Limits

When working with limits, it鈥檚 often useful to simplify expressions to make the problem more tractable. In our problem, we started by expanding the square root and then factored out \( x^2 \) to isolate it from the rest of the terms. This step is crucial for determining how the function behaves as \( x \) gets very large. Simplification may involve factoring, expanding, or canceling terms, and is a vital skill when trying to evaluate limits. Students are advised to practice manipulation of algebraic expressions as it's a recurring technique in calculus.
Limit Evaluation Techniques

Approaches to Solving Limits

There are many strategies to find the value of a limit, such as direct substitution, factorization, rationalization, and using special limit rules, like the one for limits at infinity. In the exercise, we used the approach of identifying terms that diminish as \( x \) increases. These smaller terms have less influence on the value of the function at infinity and are thus discounted. Mastering different limit evaluation techniques is essential as they come in handy for dealing with a variety of problems where the function's behavior isn鈥檛 immediately apparent.

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Most popular questions from this chapter

Evaluate the folloiwng (i) \(\lim _{x \rightarrow 0}\left[\lim _{n \rightarrow \infty}(\cos (x / 2) \cos (x / 4) \cos (x / 8) \ldots \ldots \ldots\right.\) \(\left.\cos \left(x / 2^{n}\right)\right]\) (ii) \(\lim _{n \rightarrow \infty} \sin ^{2 n} x\)

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