Problem 104 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \lim _{x \rightarrow \infty} \frac{\sqrt{1+9 x^{2}}+\sqrt{x^{2}-1}}{\sqrt{1+9 x^{2}-\sqrt{x^{2}-1}}} \text { \\{Ans. } \frac{4}{3} \text { \\} } $$

Step-by-Step Solution

Answer

When \(x \rightarrow \infty\), the limit of the given expression is \(\frac{4}{3}\).

Step 1: First, let's factor out the dominant term \(x^{2}\) from under both square roots in the numerator and denominator: Numerator: \(\sqrt{1+9x^{2}} = \sqrt{x^{2}(1/9 + 1)} = x\sqrt{1+1/9}\) \(\sqrt{x^{2}-1} = \sqrt{x^{2}(1-1/x^{2})} = x\sqrt{1-1/x^{2}}\) Denominator: \(\sqrt{1 + 9x^{2} - \sqrt{x^{2}-1}} = \sqrt{x^{2}\left(1/9 + 1 - \sqrt{1-1/x^{2}}\right)} = x\sqrt{1+1/9-\sqrt{1-1/x^{2}}}\) #Step 2: Simplify the expression#

Now, let's plug these expressions back into the overall expression and simplify: \(\frac{x(\sqrt{1+1/9}+\sqrt{1-1/x^{2}})}{x(\sqrt{1+1/9-\sqrt{1-1/x^{2}}})}\) Since x is in the numerator and the denominator, we can cancel out x: \(\frac{\sqrt{1+1/9}+\sqrt{1-1/x^{2}}}{\sqrt{1+1/9-\sqrt{1-1/x^{2}}}}\) #Step 3: Compute the limit#

Step 2: Now, as \( x \rightarrow \infty \), we can notice that \(1/x^{2} \rightarrow 0\). So, we can now substitute this into the expression and compute the limit: \(\lim _{x \rightarrow \infty}\frac{\sqrt{1+1/9}+\sqrt{1-1/x^{2}}}{\sqrt{1+1/9-\sqrt{1-1/x^{2}}}} = \frac{\sqrt{1+1/9}+\sqrt{1-0}}{\sqrt{1+1/9-\sqrt{1-0}}}\) This simplifies to: \(\frac{\sqrt{1+1/9}+\sqrt{1}}{\sqrt{1+1/9-\sqrt{1}}}\) #Step 4: Evaluate the expression#

Finally, we can evaluate our expression: \(\frac{\sqrt{10/9}+1}{\sqrt{10/9-1}} = \frac{1/3+3/3}{2/3} = \frac{4/3}{2/3} = \frac{4}{3}\) Therefore, the answer is \(\frac{4}{3}\).

Key Concepts

Asymptotic Behavior

When we talk about the asymptotic behavior of functions in calculus, we're really discussing how a function behaves as its input gets very large or very small—in other words, as it approaches infinity or negative infinity. In the exercise given, we're examining the limit of a complex rational expression as \( x \) approaches infinity.

This concept is crucial because it helps us understand infinite limits and behavior at extreme values:

• As \( x \) becomes very large, terms in the function that contribute less significantly to the overall value become negligible.
• This allows us to simplify functions by focusing on the dominant terms, like \( x^2 \) in our exercise.
• By considering the largest influencing terms, we predict the function’s behavior more efficiently.

Understanding the asymptotic behavior allows mathematicians and scientists to make predictions about the behavior of physical systems at extreme scales, often when direct measurement might be difficult.

Infinite Limits

In the context of our exercise, infinite limits help us find the output of a function as the input shoots off towards infinity. Here, the concern is the limit of \( \frac{\sqrt{1+9x^2} + \sqrt{x^2-1}}{\sqrt{1+9x^2 - \sqrt{x^2-1}}} \) as \( x \rightarrow \infty \).

Key ideas about approaching such limits involve:

• Identifying dominant terms that survive as others diminish due to large \( x \).
• The terms \( \sqrt{1/x^2} \) and \( -1/x^2 \) become insignificant as \( x \) gets larger since they approach zero.
• This realization enables simplification, focusing on the main contributors to the function’s value at infinity.

By reducing the expression to the significant terms, we can more easily find the limit—in this case yielding \( \frac{4}{3} \)—demonstrative of how the function balances itself out at infinity.

Square Root Simplification

Square root simplification is a powerful algebraic tool, especially when dealing with limits involving large values. In our exercise, the expression involves several square roots, each requiring simplification to assess the limit as \( x \to \infty \).

When simplifying square roots in the context of limits:

• Factor out the dominant terms from under the square root—\( x^2 \) in this case—to render the square root manageable.
• Transformations like \( \sqrt{1 + 9x^2} = x\sqrt{1 + 1/9} \) isolate and reduce the complexity of the root term.
• This process critically reduces unnecessary complexity in expressions where specific terms become negligible at limits.

By simplifying, we streamline the equation, making it easier to identify how other terms behave and contribute, ensuring a smooth calculation of limits. Square root manipulations, when done correctly, reveal the simple essence of complex mathematical expressions.