Problem 428 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If }|f(x)| \leq|x| \forall x, \text { find } \lim _{x \rightarrow 0} f(x) .\\{\text { Ans. } 0\\} $$

Step-by-Step Solution

Answer

By applying the squeeze theorem on the given inequality \(|f(x)| \leq |x|\), since \(\lim_{x \rightarrow 0} -|x| = 0\) and \(\lim_{x \rightarrow 0} |x| = 0\), it follows that \(\lim_{x \rightarrow 0} f(x) = 0\).

Step 1: Express the given condition as an inequality

We are given that \(|f(x)| \leq |x|\) for all values of x.

Step 2: Apply the squeeze theorem

To find \(\lim_{x \rightarrow 0} f(x)\), we can apply the squeeze theorem (which states that if a function is "trapped" between two other functions whose limits are the same, then the limit of the trapped function must also be the same) by finding two functions that bound \(f(x)\) as x approaches 0. Since \(|f(x)| \leq |x|\), we know that \(-|x| \leq f(x) \leq |x|\).

Step 3: Evaluate limits

Now, let's evaluate the limits of the bounding functions as x approaches 0: \(\lim_{x \rightarrow 0} -|x| = 0\) and \(\lim_{x \rightarrow 0} |x| = 0\)

Step 4: Apply the squeeze theorem

Since \(-|x| \leq f(x) \leq |x|\), and both \(\lim_{x \rightarrow 0} -|x|\) and \(\lim_{x \rightarrow 0} |x|\) are equal to 0, by the squeeze theorem, the limit of \(f(x)\) as x approaches 0 must also be equal to 0: \(\lim_{x \rightarrow 0} f(x) = 0\) This is the final answer.

Key Concepts

Limits

Limits are a foundational concept in calculus that describe the behavior of a function as its input approaches a certain value. The idea is to understand what happens to the function's output as the input gets arbitrarily close to a given point. When we say \( \lim_{x \rightarrow 0} f(x) = L \), it means that as \( x \) gets closer and closer to 0, \( f(x) \) approaches \( L \).
Understanding limits requires examining the behavior of the function near the point of interest, rather than just evaluating the function's value at that exact point. This concept is crucial for defining derivatives and integrals in calculus.
When you evaluate limits, you're looking for a trend or a pattern that the function values approach, even if the function is not explicitly defined at that point. It helps you determine how functions behave near discontinuities or points where they are not otherwise easily evaluable.

Inequalities

Inequalities are mathematical expressions involving the symbols \(<\), \(>\), \(\leq\), and \(\geq\). They are used to express a range of possibilities for a value rather than a precise one. In the context of calculus and the squeeze theorem, inequalities help to "trap" a function within known bounds.

• This means if a function \(f(x)\) is squeezed, or bounded, by two other functions whose behaviors are understood as they approach a limit, then \(f(x)\) will share that limit.
• For instance, if you understand that \(-|x| \leq f(x) \leq |x|\), you learn how \(f(x)\) behaves between these two boundaries.

Inequalities provide a way to handle uncertainty or variability in mathematical analysis. They are especially helpful when the exact form of a function is unknown or difficult to work with. They also allow you to comprehensively analyze limits and other aspects of function behavior without needing exact values.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. For any real number \(x\), it is denoted \(|x|\)\. It is always non-negative:

• If \(x\) is positive or zero, \(|x| = x\).
• If \(x\) is negative, \(|x| = -x\).

The use of absolute value in inequalities, like \(|f(x)| \leq |x|\), is important in bounding functions within certain limits. This setup is particularly effective for applying the squeeze theorem, as it helps define and understand the proximity of the function to zero.
Absolute value inequalities describe intervals around zero, giving a natural way to express uncertainty or variability without assigning a sign. This concept helps solidify the intuitive understanding of "closeness" to a number or point, especially when evaluating the behavior of functions with respect to limits.