Problem 35 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow 0} \frac{e^{x}}{x^{2}}\\{\text { Ans. } \infty\\} $$

Step-by-Step Solution

Answer

As \(x\) approaches \(0\), the denominator \(x^2\) goes to \(0\), while the numerator \(e^x\) goes to \(1\). Since \(e^x\) decreases slower than \(x^2\) for negative \(x\) and increases faster than \(x^2\) for positive \(x\), the ratio \(\frac{e^x}{x^2}\) increases without bound as \(x\) gets closer to \(0\). Thus, the limit of the function is infinite: \( \lim_{x \rightarrow 0} \frac{e^x}{x^2} = \infty \).

Step 1: Identify the form of the limit

The limit is given by \( \lim_{x \rightarrow 0} \frac{e^x}{x^2} \). We need to find the value of this limit as \( x \) approaches \( 0 \).

Step 2: Investigate the behavior of the function as x approaches 0

As \(x\) approaches \(0\), we can see that the denominator \(x^2\) goes to \(0\), and the numerator \(e^x\) goes to \(1\), since \(e^0 = 1\). Therefore, the function behaves similar to a \(\frac{1}{0}\) form, but we need to carefully analyze the ratio's behavior.

Step 3: Compare the rates at which the numerator and denominator approach their respective limits

As \(x\) approaches \(0\), \(e^x\) approaches \(1\) and \(x^2\) approaches \(0\). We have the following behavior: - \(e^x\) decreases slower than \(x^2\) for negative \(x\). - \(e^x\) increases faster than \(x^2\) for positive \(x\). Thus, it seems that the ratio \(\frac{e^x}{x^2}\) would increase without bound for both negative and positive \(x\) approaching \(0\).

Step 4: Determine the limit

Based on the rate comparison in the previous step, we can conclude that as \(x\) approaches \(0\), the ratio \(\frac{e^x}{x^2}\) increases without bound. Thus, the limit of the function is infinite: \( \lim_{x \rightarrow 0} \frac{e^x}{x^2} = \infty \).

Key Concepts

Exponential Functions

Exponential functions, notably in the form of \( f(x) = e^x \), are a central concept in mathematics because of their unique properties. The base of the exponential function, \( e \), is known as Euler’s number, approximately 2.718. It's a fundamental constant in mathematics.

• Exponential functions grow rapidly as the variable increases.
• They are continuous and differentiable everywhere.
• The derivative of \( e^x \) is easy to work with, as it remains \( e^x \).

Most importantly, exponential functions have a property that makes them very smooth; they are neither zero nor negative. This makes exponential functions quite interesting when analyzing limits, particularly because their growth can dominate many other functions.

Behavior of Limits

The behavior of limits is crucial when evaluating how a function behaves as it approaches a particular value, especially infinity or zero. Limits help us understand the end behavior of functions and their compositions.For example, consider the function \( \frac{e^x}{x^2} \). To understand its limit as \( x \) approaches 0, we study the individual behaviors of the numerator and the denominator:

• The function \( e^x \) approaches 1 as \( x \to 0 \) because \( e^0 = 1 \).
• The denominator \( x^2 \) approaches 0, creating a challenge because dividing by zero typically causes an undefined value.

This results in a distinctive behavior around \( x = 0 \). The effect of these behaviors combined is an infinite growth in the function because the rate of change in the numerator and denominator determines how they relate as they approach their limits. Understanding these behaviors is key to knowing when and why a limit might become infinite.

Infinite Limits

An infinite limit occurs when a function's value grows without bound as the independent variable approaches a certain point, such as in the function \( \frac{e^x}{x^2} \) as \( x \to 0 \). This concept explains situations where limits do not converge to a real number.In our given exercise:

• As the denominator \( x^2 \) approaches 0, it suggests the potential for division by zero.
• The numerator \( e^x \) remains stable, near 1, yet grows to dominate the small values of \( x^2 \).
• This discrepancy causes the function to increase indefinitely, resulting in an infinite limit.

The recognition of infinite behavior is essential when distinguishing between merely large outputs and truly infinite limits. Infinite limits often require specific mathematical tools and reasoning to fully analyze since they describe the unbounded nature of a function's growth at certain points.