Problem 38 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow \infty} x+e^{x} \quad\\{\text { Ans. } \infty\\} $$

Step-by-Step Solution

Answer

The short answer to the question is: As x approaches infinity, both the linear term 'x' and the exponential term \(e^x\) increase without bound. Therefore, their sum will also approach infinity. So, \(\lim _{x \rightarrow \infty} (x + e^x) = \infty\).

Step 1: Identify the terms of the expression

The given expression consists of two terms: \(x\) and \(e^x\). We need to analyze how each of these terms behaves as x approaches infinity.

Step 2: Analyze the behavior of the first term 'x' as x approaches infinity

The first term is simply 'x', which is a linear function. As x approaches infinity, the value of 'x' keeps increasing without bound. Thus, the first term also approaches infinity when x approaches infinity.

Step 3: Analyze the behavior of the second term 'e^x' as x approaches infinity

The second term is the exponential function \(e^x\), which grows very rapidly as x increases. It is well-known that the exponential function grows much faster than any linear function. Thus, as x approaches infinity, the second term also approaches infinity.

Step 4: Combine the behavior of both terms to determine the overall behavior

Since both the terms (x and \(e^x\)) approach infinity as x approaches infinity, their sum will also approach infinity. Therefore, the limit of the given expression as x approaches infinity is also infinity. So, we have: \(\lim _{x \rightarrow \infty} (x + e^x) = \infty\)

Key Concepts

Exponential growth

Exponential growth describes a process where the rate of change of a quantity is proportional to the current value of that quantity. This means that as a variable, like our x in the example, increases, the quantity increases so rapidly that it seems to shoot up vertically in a graph. The term \( e^x \) is an example of exponential growth. Here is why exponential growth is so powerful:

• Rapid Increase: As x becomes larger, \( e^x \) increases faster than linear growth.
• Dominating Term: When comparing exponential functions to polynomial or linear functions, the exponential will eventually become much larger.

Here, as x approaches infinity, \( e^x \) grows enormously, eclipsing any linear component like \( x \) in the equation. This powerful growth characteristic means that the exponential term heavily influences the outcome of limits involving sums with exponential terms.

Linear function

A linear function is a basic function in mathematics with a graph that resembles a straight line. A linear function has the general form \( y = mx + c \), where m is the slope, and c is the y-intercept. In this problem, the term x indicates a linear function:

• Constant Growth Rate: Unlike the rapid increase in exponential growth, linear functions grow steadily with a constant rate.
• Simplicity: Linear functions are easy to model and understand, making them foundational in mathematics.

As x gets very large, the value of x also trends toward infinity, but not at the same explosive rate as an exponential function like \( e^x \). Even though the linear function increases without bound as x approaches infinity, it does so less dramatically. Understanding linear functions helps in analyzing other types of growth, serving as a contrast to the more powerful exponential growth.

Behavior at infinity

When discussing limits and growth rates, understanding what happens "at infinity" is essential. The phrase describes the behavior of a function or an expression as the variable within it approaches infinity (becomes indefinitely large). In the given expression \( x + e^x \):

• Summation Influence: Since both parts of the expression tend toward infinity, their sum naturally follows.
• Limit Analysis: In limit problems, dominant terms dictate the outcome. Here, \( e^x \) will be the dominating term.

At infinity, exponential terms typically overshadow linear parts due to their rapid growth. Hence, when determining the behavior of an expression like the one given, the limit tends towards the same direction influenced by the fastest-growing term. Recognizing which terms influence behavior at infinity ensures you can handle a range of limit problems successfully.