Problem 437 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow+\infty} \frac{x^{b}}{a^{x}} $$

Step-by-Step Solution

Answer

The short answer for the question is: \(\lim_{x \rightarrow +\infty} \frac{x^b}{a^x} = 0\), because exponential functions (\(a^x\)) grow faster than power functions (\(x^b\)) as x approaches infinity, causing the denominator to become much larger than the numerator.

Step 1: Analyze the functions

The given limit has two functions: 1. \(x^b\) in the numerator – a power function, where b is a constant 2. \(a^x\) in the denominator – an exponential function, where a is a constant

Step 2: Compare the growths of the functions

As x approaches positive infinity, we want to compare the growth of both functions. - Power functions (\(x^b\)): As x increases, the value of the function increases as well at a rate determined by the constant b. - Exponential functions (\(a^x\)): As x increases, the value of the function increases rapidly, with a growth rate determined by the constant a. In general, exponential functions grow faster than power functions as x approaches infinity.

Step 3: Calculate the limit

Using the information from Step 2, we know that the exponential function in the denominator will grow faster than the power function in the numerator. This means that the denominator will become much larger than the numerator as x approaches infinity. As a result, the fraction goes to zero. Mathematically, we can write this as: \[ \lim_{x \rightarrow +\infty} \frac{x^b}{a^x} = 0 \]

Step 4: Present the solution

The limit of the given function as x approaches positive infinity is 0, expressed as: \[ \lim_{x \rightarrow +\infty} \frac{x^b}{a^x} = 0 \]

Key Concepts

Power Function

Power functions are expressions of the form \( x^b \) where \( b \) is a constant. These functions represent a number \( x \) being raised to a power, indicated by \( b \). The behavior of power functions is primarily determined by the value of \( b \). When \( b \) is a positive integer, as \( x \) grows larger, \( x^b \) increases quite rapidly but at a consistent rate.

• For example, if \( x^b = x^2 \), doubling \( x \) results in quadrupling \( x^2 \).
• If \( b \) is a negative integer, the function represents a reciprocal power, such as \( x^{-2} = \frac{1}{x^2} \), which decreases as \( x \) grows.
• For fractional powers, like \( x^{\frac{1}{2}} = \sqrt{x} \), the function increases at a slower rate.

Power functions are useful in modeling various phenomena, from the speed of a freely falling object to the intensity of a light beam. Their growth, although significant, is surpassed by exponential functions as \( x \) approaches infinity.

Exponential Function

An exponential function is expressed in the form \( a^x \), where \( a \) is a positive constant base and \( x \) is the exponent. Unlike power functions, the variable \( x \) is the exponent, which leads to much faster growth.

• For example, with \( a = 2 \), the function \( 2^x \) means the number 2 is being multiplied by itself \( x \) times.
• This rapid multiplication results in exponential growth, which can far outpace the growth of power functions.
• The larger the base \( a \), the faster the exponential function will grow.

Exponential functions are crucial in fields ranging from finance (compounding interest) to biology (population growth and decay). Their rapid growth makes them dominate over power functions when \( x \) becomes very large. In the context of limits, this behavior explains why \( \frac{x^b}{a^x} \) tends towards zero as \( x \) approaches infinity.

Asymptotic Behavior

Asymptotic behavior describes how functions behave as the input \( x \) approaches a specific value, usually infinity. In the context of limits, understanding this behavior is key to predicting how a ratio of functions will behave.

• When considering \( \lim_{x \rightarrow +\infty} \frac{x^b}{a^x} \), the exponential function \( a^x \) in the denominator grows so much faster than the power function \( x^b \) in the numerator that the fraction becomes negligibly small.
• This rapid dominance results in the literal 'flattening out' of the function value towards zero.
• In practical terms, asymptotic behavior helps us understand limits and can be useful in approximating complex calculations.

So, when determining limits involving power and exponential functions, recognizing the asymptotic behavior of these functions allows us to solve the problem efficiently. For \( \frac{x^b}{a^x} \) as \( x \rightarrow +\infty \), the exponential growth leads the fraction to approach zero.