Problem 78 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\lim _{x \rightarrow 0}\left[\frac{a \sin x}{x}\right]+\left[\frac{b \tan x}{x}\right]\), where \(a, b\) are integers and [] denotes integral part, is equal to (A) \(a+b\) (B) \(a+\underline{b-1}\) (C) \(a-b\) (D) \(a-\underline{b-1}\)

Step-by-Step Solution

Answer

(A) a + b

Step 1: Understand Limit Property

For small values of \(x\), we have the limit property \(\lim_{x \to 0} \frac{\sin x}{x} = 1\) and \(\lim_{x \to 0} \frac{\tan x}{x} = 1\). This is because the sine and tangent functions are approximately equal to \(x\) when \(x\) is close to zero.

Step 2: Apply the Limit Property

Using the limit properties from Step 1, for the given expression, we have \(\lim_{x \to 0} \frac{a \sin x}{x} = a \cdot 1 = a\) and \(\lim_{x \to 0} \frac{b \tan x}{x} = b \cdot 1 = b\).

Step 3: Analyze the Integral Part

The expression given is in integer form, denoted by \([]\). For a limit that reaches exactly an integer, the integral part remains the integer. Hence, the integral of \(a\) is \(a\) and \(b\) is \(b\).

Step 4: Combine the Results

Adding the integral parts gives \(a + b\). Thus, we check the answer choices: (A) \(a + b\), (B) \(a + b - 1\) underlines \(b-1\), (C) \(a - b\), (D) \(a - b + 1\). Choice (A) matches our computed value.

Key Concepts

Sine Limit Property

The sine limit property is a well-known concept in calculus that simplifies certain trigonometric limits. When examining the function \[ \lim_{x \to 0} \frac{\sin x}{x}, \] we find that as \(x\) approaches zero, the value approaches 1. This property becomes useful in solving problems involving limits, especially when the sine function is involved. Here's why:

• For very small \(x\), \(\sin x\) is closely approximated by \(x\).
• Thus, the division \(\frac{\sin x}{x}\) yields a result that approaches 1 as \(x\) nears zero.

This relationship allows for simplification in both theoretical applications and practical calculations where precision around zero is necessary. It is essential for effectively solving limit problems that involve sine functions.

Tangent Limit Property

Similar to the sine limit property, the tangent limit property is another simplifying tool in calculus. When analyzing:\[ \lim_{x \to 0} \frac{\tan x}{x}, \] an interesting thing happens. The result tends towards 1 as well. This is because:

• For small values of \(x\), \(\tan x\) is nearly equivalent to \(x\).
• This results in the expression \(\frac{\tan x}{x}\) approaching 1.

This property is extremely useful in computations and proofs where precision around zero is essential. It simplifies the process of solving limits involving tangent and helps provide clear and correct solutions without complicated calculations.

Integral Part of Function

The concept of the integral part of a function, often denoted by square brackets \([]\), refers to finding the greatest integer less than or equal to a given number. This is also known as the floor function. In mathematical notation, the integral part of \(x\) is expressed as \(\lfloor x \rfloor\). Some key points:

• The integral part is crucial in analyzing whole number results from limits and sums in calculus.
• For positive numbers, it simply "drops" any decimal portion, thereby ensuring the result is a whole number.
• In the context of limits, when the expression evaluates to exactly an integer, the integral part doesn't change.

In the given exercise, the integral part ensures that when applying the properties for sine and tangent limits, the results are maintained as integers \(a\) and \(b\). This ultimately helps in determining the correct sum, which in this case is \(a + b\).