Problem 2 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f(x)=\sin x, \quad x \neq n \pi\) \(=2, \quad x=n \pi\) where \(n \in Z\) and \(g(x)=x^{2}+1, \quad x \neq 2\) \(=3\), \(x=2\). then \(\lim _{x \rightarrow 0} g[f(x)]\) is (A) 1 (B) 0 (C) 3 (D) Does not exist

Step-by-Step Solution

Answer

The limit \( \lim_{x \to 0} g[f(x)] \) is 1.

Step 1: Understanding the Function Composition

We are asked to find the limit of the composition of two functions, specifically \( g[f(x)] \), as \( x \) approaches 0. The function \( f(x) \) is defined piecewise, and we need to determine its behavior near \( x = 0 \). Since \( 0 eq n\pi \), we use \( f(x) = \sin x \) near \( x = 0 \). The function \( g(x) \) is \( x^2 + 1 \) except at \( x = 2 \), which doesn’t affect us as \( \sin x \) for \( x \) near zero is not equal to 2.

Step 2: Evaluate \( f(x) \) Near 0

Since \( f(x) = \sin x \) when \( x eq n\pi \) and \( 0 eq n\pi \), near \( x = 0 \), we have \( f(x) = \sin x \). As \( x \to 0 \), \( \sin x \to \sin(0) = 0 \). Therefore, \( f(x) \to 0 \) as \( x \to 0 \).

Step 3: Substitute \( f(x) \) into \( g(x) \)

We now need to find \( g(f(x)) \) as \( \lim_{x \to 0} g[f(x)] = \lim_{x \to 0} g[\sin x] = g[\lim_{x \to 0} \sin x] = g(0) \). Since \( g(x) = x^2 + 1 \) when \( x eq 2 \), we find \( g(0) = 0^2 + 1 = 1 \).

Step 4: Evaluating the Limit

With all substitutions and evaluations done, the limit becomes \( \lim_{x \to 0} g[f(x)] = 1 \). Thus, the value of \( \lim_{x \to 0} g[f(x)] \) is 1.

Key Concepts

Piecewise functions

A piecewise function is a function defined by different expressions depending on the value of the input. Imagine a route that changes depending on where you are. It's a bit like a mathematical GPS!

In our exercise, the function \( f(x) \) is defined in two parts. It equals \( \sin x \) for values of \( x \) that are not multiples of \( \pi \), and it takes the constant value of 2 when \( x \) is a multiple of \( \pi \). This flexibility allows the function to behave differently depending on the input. The function \( g(x) \) follows a similar pattern, offering two different expressions depending on whether \( x \) is 2 or not.

To solve limits involving piecewise functions, we must carefully choose the expression that describes the function at the point of interest, in this case as \( x \to 0 \). It's essential to know which part of the function applies at critical values, helping avoid wrong turns in our calculations.

Function composition

Function composition involves applying one function to the results of another, like a two-step recipe. If you think about making a sandwich, one function gives you the bread, and the other adds the filling.

In mathematics, when you have two functions \( f(x) \) and \( g(x) \), the composite function is represented as \( g(f(x)) \). This means you first apply \( f(x) \), and then use that result as the input for \( g(x) \). This chain reaction is what we're solving in the exercise as \( g[f(x)] \).

To evaluate this expression as \( x \to 0 \), we first find how \( f(x) \) behaves at this point. Here, since \( f(x) = \sin x \) for values near zero, \( f(x) \to 0 \). Hence, the next step is substituting this value into \( g(x) \). With \( g(x) = x^2 + 1 \), it becomes \( g(0) \), which simplifies our journey to finding the limit.

Trigonometric functions

Trigonometric functions like sine, cosine, and tangent, are foundational in mathematics, especially in calculus and geometry. These functions relate angles to ratios of sides in right triangles and have periodic properties.

In the exercise, we're working with the sine function, \( \sin x \). For small values of \( x \), such as those near zero, the sine function behaves in a very predictable way: \( \sin x \to 0 \) as \( x \to 0 \). This behavior is crucial when evaluating the limit in our problem, as it directly informs the value of \( f(x) \) used in the function composition.

Understanding these fundamental trigonometric behaviors allows us to solve complex problems more easily, especially when combined with other mathematical concepts like limits and compositions. It's like having a trusty compass when navigating through a dense forest of equations.