Problem 8 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The value of \(\lim _{x \rightarrow \infty} \frac{x^{5}}{5^{x}}\) is (A) 1 (B) \(-1\) (C) 0 (D) None of these

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Function

We have the function \( \frac{x^5}{5^x} \) whose limit we are evaluating as \( x \rightarrow \infty \). It is a rational function with the polynomial term \( x^5 \) in the numerator and an exponential term \( 5^x \) in the denominator.

Step 2: Consider the Behavior of the Terms

As \( x \) approaches infinity, the polynomial \( x^5 \) grows at a much slower rate than the exponential function \( 5^x \). Exponential functions dominate polynomials as \( x \) becomes very large.

Step 3: Apply Limits to Polynomial and Exponential Terms

The exponential \( 5^x \) grows substantially faster than the polynomial \( x^5 \), so the fraction \( \frac{x^5}{5^x} \) approaches zero as \( x \to \infty \).

Step 4: Conclude the Limit

Since the exponential function in the denominator grows significantly more than the polynomial in the numerator, the fraction effectively becomes zero as \( x \to \infty \). Thus, \( \lim_{x \to \infty} \frac{x^5}{5^x} = 0 \).

Key Concepts

Understanding Polynomial Growth

Polynomial growth refers to how the terms in a polynomial, like the polynomial in our function, grow as the variable increases. A polynomial is a mathematical expression consisting of variables raised to powers and combined with coefficients, such as \( x^5 \). The maximum power of the variable, which is 5 in this case, dictates how fast the polynomial grows.

• For example, if you increase \( x \), a term like \( x^5 \) grows quickly, but not at a rate beyond what its exponent dictates.
• This growth rate for polynomials is relatively consistent and predictable.

As \( x \) becomes very large, the polynomial terms dominate lower-order terms but are still outpaced by exponential terms, which we will explore further in the next section.

The Nature of Exponential Growth

Exponential growth, involving terms like \( 5^x \) in our function, occurs when quantities grow at rates proportional to their current value. This rapid increase is characteristic of exponential functions.

• For instance, each increment in \( x \) results in \( 5^x \) increasing by a factor of 5, which is why it becomes very large very quickly.
• Exponential growth is often associated with phenomena such as population growth, radioactive decay, and compound interest.

In mathematical terms, exponential growth always outpaces polynomial growth as \( x \to \infty \). This is why, in our limit exercise, \( 5^x \) in the denominator grows faster than \( x^5 \) in the numerator, leading to the conclusion that \( \lim_{x \to \infty} \frac{x^5}{5^x} = 0 \).

Exploring Rational Functions

Rational functions are expressions formed by dividing two polynomials. In the given exercise, we have \( \frac{x^5}{5^x} \), which is a quotient of a polynomial function by an exponential function.

• Generally, the behavior of rational functions as \( x \to \infty \) is examined by looking at the degree of the polynomials in the numerator and denominator.
• However, in cases like our exercise where you have a polynomial over an exponential, the exponential function dictates the behavior.

When dealing with limits, understanding which part of the rational function grows faster helps predict its behavior as \( x \to \infty \).Since \( 5^x \) grows much faster than \( x^5 \), the whole expression \( \frac{x^5}{5^x} \) shrinks toward zero. Thus, recognizing and analyzing the growth rates of the components of rational functions is key to determining their limits.