Problem 55 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\begin{aligned} &\text { Given }\\\ &\begin{aligned} f(x) &=x, \quad x<0 \\ &=1, \quad x=0 \\ &=x^{2}, \quad x>0 \\ \text { Does } & \lim _{x \rightarrow 0} f(x) \text { exist? \\{ns. Yes\\} } \end{aligned} \end{aligned}

Step-by-Step Solution

Answer

The limit of the given piecewise function \(f(x)\) as \(x\) approaches 0 exists, and it is \(\lim_{x \rightarrow 0} f(x) = 0\).

Step 1: Determine the left-hand limit

As x approaches 0 from the left (x < 0), the function f(x) is given by f(x) = x. So, we have to find the limit of f(x) = x as x approaches 0. Using substitution, we get: \[\lim_{x \rightarrow 0^{-}} x = 0\]

Step 2: Determine the right-hand limit

As x approaches 0 from the right (x > 0), the function f(x) is given by f(x) = x^2. So, we have to find the limit of f(x) = x^2 as x approaches 0. Using substitution, we get: \[\lim_{x \rightarrow 0^{+}} x^2 = 0^2 = 0\]

Step 3: Compare the left-hand and right-hand limits

Now that we have both limits (from the left and right side), we can determine if the limit of f(x) exists by comparing these limits: \[\lim_{x \rightarrow 0^{-}} f(x) = \lim_{x \rightarrow 0^{+}} f(x) = 0\] As both the left-hand and right-hand limits are equal, the limit of f(x) as x approaches 0 exists.

Step 4: State the final answer

The limit of f(x) as x approaches 0 exists, and it is: \[\lim_{x \rightarrow 0} f(x) = 0\]

Key Concepts

Left-Hand Limit

When we talk about the left-hand limit of a function, we are interested in what happens to the function as the input value approaches a certain point from the negative, or left, side. Think of it as sneaking up on the number from smaller numbers. In our example, we are interested in what happens as we approach 0 from the left, which is why we write

• \( \lim_{x \rightarrow 0^{-}} f(x) \)

This notation specifically refers to the values of \( x \) when \( x < 0 \). For the exercise provided, the function behaves as \( f(x) = x \) for values less than 0. Therefore, as \( x \) approaches 0 from the left-hand side, the value of \( f(x) \) approaches 0 as well.The calculation becomes quite straightforward:

• \( \lim_{x \rightarrow 0^{-}} x = 0 \)

The left-hand limit evaluates to 0, meaning that as we get closer and closer to 0 from the left, the function’s output gets closer and closer to 0.

Right-Hand Limit

The right-hand limit is essentially the opposite of the left-hand limit. It focuses on what happens as the input value approaches a certain point from the positive, or right, side. To imagine this, think about getting closer to the number from larger numbers. The notation we use for this is:

• \( \lim_{x \rightarrow 0^{+}} f(x) \)

This is specifically considering the values of \( x \) when \( x > 0 \). In our exercise, the function is defined as \( f(x) = x^2 \) for values greater than 0. Therefore, as \( x \) gets closer to 0 from the right-hand side, the value of \( f(x) \) also moves toward 0.To calculate it:

• \( \lim_{x \rightarrow 0^{+}} x^2 = 0^2 = 0 \)

The right-hand limit is found to be 0, signifying that as we inch closer to 0 from the right, the output of the function similarly approaches 0.

Existence of a Limit

The concept of a limit's existence hinges on the behavior of both the left-hand and right-hand limits. A limit at a particular point exists only if the left-hand and right-hand limits are equal. This means that the function's behavior from both sides is heading towards the same value.In this exercise, we have:

• The left-hand limit: \( \lim_{x \rightarrow 0^{-}} f(x) = 0 \)
• The right-hand limit: \( \lim_{x \rightarrow 0^{+}} f(x) = 0 \)

Since both these limits are equal, we conclude that:

• \( \lim_{x \rightarrow 0} f(x) = 0 \)

Thus, the limit of \( f(x) \) as \( x \) approaches 0 exists and is equal to 0.Understanding the existence of limits is crucial. It tells us that, at this point, the function behaves predictably from either direction. When both directional limits meet at one value, it means that there is no sudden change or discontinuity in the function at that point.