Problem 302 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow \infty} \cos \sqrt{x+1}-\cos \sqrt{x} \quad\\{\text { Ans. } 0\\} $$

Step-by-Step Solution

Answer

The limit of the given function, \(\lim_{x \rightarrow \infty} \cos \sqrt{x+1} - \cos \sqrt{x}\), can be found by rewriting it as \(\cos( \sqrt{x+1} - \sqrt{x})\). As \(x\) goes to infinity, the angle inside the cosine function approaches 0, and thus the limit of the original function approaches \(\cos(0) = 1\), making the result \(\lim_{x \rightarrow \infty} \cos \sqrt{x+1} - \cos \sqrt{x} = 0\).

Step 1: Rewrite the expression using angle subtraction formula for cosine

We have the expression \(\cos \sqrt{x+1} - \cos \sqrt{x}\), and we can rewrite it using the angle subtraction formula for the cosine function: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \] where \(A = \sqrt{x+1}\) and \(B = \sqrt{x}\). Then, the given expression could be written as: \[ \cos( \sqrt{x+1} - \sqrt{x}) \]

Step 2: Analyze the new expression when x approaches infinity

Now let's analyze the behavior of \(\cos( \sqrt{x+1} - \sqrt{x})\) as \(x\) approaches infinity. Note that when \(x \rightarrow \infty\), the value of the difference inside the cosine function approaches zero, which can be shown as: \[ \lim_{x \rightarrow \infty} (\sqrt{x+1} - \sqrt{x}) \] Using a little algebra, we can rewrite the above expression as: \[ \lim_{x \rightarrow \infty} \frac{1}{\sqrt{x+1} + \sqrt{x}} \] Since \(x\) is going to infinity, the expression inside the limit approaches 0: \[ \lim_{x \rightarrow \infty} \frac{1}{\sqrt{x+1} + \sqrt{x}} = 0 \]

Step 3: Determine the limit of the function

Since the limit we found in the previous step shows that the difference of the angles inside the cosine function is approaching 0 as \(x\) goes to infinity, we can apply this result to find the limit of the original function: \[ \lim_{x \rightarrow \infty} \cos( \sqrt{x+1} - \sqrt{x}) = \cos(0) \] We know that \(\cos(0) = 1\), therefore: \[ \lim_{x \rightarrow \infty} \cos \sqrt{x+1} - \cos \sqrt{x} = 0 \] So, the limit of the given function as \(x\) goes to infinity is 0.

Key Concepts

Cosine Angle Subtraction Formula

Understanding the cosine angle subtraction formula is vital for solving certain types of trigonometric problems. At its core, this formula expresses the cosine of the difference between two angles in terms of the cosines and sines of the angles themselves. The generic formula is:

\[ \text{cos}(A - B) = \text{cos} A \times \text{cos} B + \text{sin} A \times \text{sin} B \
\] When applied to an expression like \( \text{cos} \sqrt{x+1} - \text{cos} \sqrt{x} \), one might be tempted to simply subtract the values. However, using the angle subtraction formula allows for greater manipulation and facilitates finding the limit. In the given example, \(A = \sqrt{x+1}\) and \(B = \sqrt{x}\). The aim is to express the initial complex expression in a form that makes it easier to evaluate its limit as \(x\) approaches infinity.

Behavior of Limits at Infinity

When dealing with limits at infinity, we are essentially interested in the behavior of a function as the input grows without bound. For the cosine function, as with many trigonometric functions, this can be particularly intriguing due to periodicity. However, in some cases, the inside of the cosine function can approach a number as \(x\) goes to infinity.

For instance, the difference \(\sqrt{x+1} - \sqrt{x}\) appears to approach zero as \(x\) becomes very large. By manipulating the expression algebraically, we can show that its limit is indeed zero. This tells us that the entire expression, \(\text{cos}(\sqrt{x+1} - \sqrt{x})\), will approach \(\text{cos}(0)\), which is a known value. By understanding how limits behave at infinity, we can solve limits that may initially seem daunting due to their complex appearance.

Algebraic Manipulation of Limits

Algebraic manipulation is a foundational technique in calculating limits. It involves rewriting expressions in a form that makes taking their limits straightforward. In the step-by-step solution presented, a critical algebraic trick is to rationalize the difference \(\sqrt{x+1} - \sqrt{x}\). By recognizing the form of \(a^2 - b^2 = (a+b)(a-b)\), we can convert the numerator to 1—a constant—and the denominator to a sum of roots that simplifies the expression.

Upon this simplification, the result in the denominator becomes larger as \(x\) increases, making it formidable to deduce that the whole fraction approaches zero. Mastery of these algebraic maneuvers can make a significant difference in solving limits, and is a skill that is essential to the toolkit of anyone tackling calculus problems.