Question

Evaluate the following limits: \(\lim _{x=\infty} \frac{a x+b x^{-1}}{c x+d x^{-1}}\)

Short answer

Question: Find the limit of the given expression as x approaches infinity: $$\lim _{x=\infty} \frac{a x+b x^{-1}}{c x+d x^{-1}}$$ Answer: The limit of the expression as x approaches infinity is given by: $$\lim _{x=\infty} \frac{a x+b x^{-1}}{c x+d x^{-1}} = \frac{a}{c}$$

Step-by-step solution

Identify the highest power of x in the denominator

In this case, the highest power of x in the denominator is x. Therefore, we will divide both the numerator and the denominator by x. #Step 2: Divide numerator and denominator by the highest power of x#

Divide numerator and denominator by x

Divide both the numerator and the denominator by x: $$\lim _{x=\infty} \frac{a x+b x^{-1}}{c x+d x^{-1}} = \lim _{x=\infty} \frac{a x/x+b x^{-1}/x}{c x/x+d x^{-1}/x}$$ #Step 3: Simplify the expression#

Simplify the expression

Simplify the expression obtained in Step 2: $$\lim _{x=\infty} \frac{a x+b x^{-1}}{c x+d x^{-1}} = \lim _{x=\infty} \frac{a+b x^{-2}}{c+d x^{-2}}$$ #Step 4: Evaluate the limit as x approaches infinity#

Evaluate the limit as x approaches infinity

Now we need to evaluate the limit: As x approaches infinity, the terms with x in the denominator (i.e., \(b x^{-2}\) and \(d x^{-2}\)) approach zero. So, the simplified expression becomes: $$\lim _{x=\infty} \frac{a+b x^{-2}}{c+d x^{-2}} = \lim _{x=\infty} \frac{a}{c}$$ Since a and c are constants, the limit is just the ratio of the constants: $$\lim _{x=\infty} \frac{a x+b x^{-1}}{c x+d x^{-1}} = \frac{a}{c}$$

Key concepts

Understanding Limits in Calculus

Limits are a fundamental concept in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of this exercise, we evaluate limits as the variable \(x\) approaches infinity. This implies we're interested in the behavior of the function as \(x\) becomes very large.

When calculating limits for rational functions as \(x\) approaches infinity, we often simplify the function by dividing by the highest power of \(x\) in the denominator. This step reveals which terms in the expression become negligible. These negligible terms usually involve negative powers of \(x\), or fractions with \(x\) in the denominator, which approach zero as \(x\) increases.

Understanding limits helps us identify long-term behavior of functions in calculus, which is crucial in fields such as physics, engineering, and economics. They provide insights into function behavior without evaluating the function at the point of interest directly.

Exploring Rational Functions

Rational functions, in terms of calculus, are expressions formed as the ratio of two polynomials. For example, the given problem is a rational function with both the numerator and the denominator being linear combinations of \(x\) and \(x^{-1}\).

• Numerator: \(a x + b x^{-1}\)
• Denominator: \(c x + d x^{-1}\)

When working with rational functions, especially as \(x\) approaches infinity, we are interested in how the highest power of \(x\) affects the output. This is because higher powers of \(x\) will grow more significantly than lower powers, meaning they initially determine the behavior of the function.

Rational functions are significant in calculus because they can approximate other functions quite well and often describe real-world processes. Simplifying them by removing insignificant terms when \(x\) becomes very large can help solve limits easily.

Investigating Asymptotic Behavior

Asymptotic behavior in the context of calculus and rational functions refers to the behavior of a function as it approaches infinity or a particular value. For rational functions like in the problem, as \(x\) becomes very large, the function approaches the value of the ratio of the leading coefficients of the numerator and the denominator. In this case, it simplifies to \(\frac{a}{c}\).

Understanding asymptotic behavior is important because it gives a predictive insight into the behavior of the function at extreme values or limits. Recognizing that parts of the function become negligible allows for simplifications, like reducing the expression to its most significant terms.

• Asymptotes represent the lines that the graph of the function approaches but never touches or crosses completely.
• This concept in calculus is akin to predicting the long-term behavior or extreme conditions of systems described by functions.

In conclusion, asymptotic behavior provides essential tools for simplifying functions at limits, enhancing our understanding of the broader behavior of rational functions.