Problem 177 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \left.\lim _{x \rightarrow \pm \infty} \sqrt{9 x^{2}+1}-3 x \text { \\{Ans. } 0,+\infty\right\\} $$

Step-by-Step Solution

Answer

The limits of the function \(f(x) = \sqrt{9x^2 + 1} - 3x\) as \(x\) approaches positive and negative infinity are: \(\lim_{x\to +\infty}f(x) = 0\) \(\lim_{x\to -\infty}f(x) = +\infty\) Thus, the answer is \(\{0, +\infty\}\).

Step 1: Rewrite the function

We can simplify our work by first rewriting the given function in a more suitable form. Let \(f(x) = \sqrt{9x^2 + 1} - 3x\).

Step 2: Evaluate limits separately

Let's find the limit of the function as \(x\) approaches positive infinity, and negative infinity separately.

Step 3: Find limit as \(x\) approaches positive infinity

To find \(\lim_{x\to +\infty}f(x)\), we can use the following trick: \[ \begin{aligned} \lim_{x\to +\infty}f(x) &= \lim_{x\to +\infty}\left(\sqrt{9x^2 + 1} - 3x\right) \\ & = \lim_{x\to +\infty} \frac{\left(\sqrt{9x^2 + 1} - 3x\right)\left(\sqrt{9x^2 + 1} + 3x\right)}{\left(\sqrt{9x^2 + 1} + 3x\right)} \\ &=\lim_{x\to +\infty}\frac{9x^2 + 1 - 9x^2}{\sqrt{9x^2 + 1} + 3x} \\ &=\lim_{x\to +\infty}\frac{1}{\sqrt{9x^2 + 1} + 3x}\\ \end{aligned} \] Since \(9x^2 + 1\) and \(3x\) both approach infinity as \(x\) approaches positive infinity, the expression \(\frac{1}{\sqrt{9x^2 + 1} + 3x}\) approaches 0. Thus, \(\lim_{x\to +\infty}f(x) = 0\).

Step 4: Find limit as \(x\) approaches negative infinity

Next, we'll find \(\lim_{x\to -\infty}f(x)\): \[ \begin{aligned} \lim_{x\to -\infty}f(x) &= \lim_{x\to -\infty}\left(\sqrt{9x^2 + 1} - 3x\right) \\ \end{aligned} \] Since \(9x^2 + 1\) approaches infinity as \(x\) approaches negative infinity, the expression \(\sqrt{9x^2 + 1}\) also approaches infinity. However, \(3x\) approaches negative infinity as \(x\) approaches negative infinity. Thus, \(\lim_{x\to -\infty}f(x) = +\infty\).

Step 5: Combine the results

We have found the limits of the function as \(x\) approaches positive infinity and negative infinity: \(\lim_{x\to +\infty}f(x) = 0\) \(\lim_{x\to -\infty}f(x) = +\infty\) So the final answer is \(\{0, +\infty\}\).

Key Concepts

Asymptotic Behavior

Understanding the concept of asymptotic behavior is key when exploring limits at infinity. It involves analyzing how a function behaves as its input, or x, moves towards infinity or negative infinity. In simpler terms, it's like peering into a function's future and seeing how it stabilizes (or doesn't) as the values get extremely large.

Considering our example, \( \lim_{x\to +\infty}f(x) \) and \( \lim_{x\to -\infty}f(x) \) describe the behavior of the function \( f(x) = \sqrt{9x^2 + 1} - 3x \) as x becomes very large or very small. We found that as x approaches positive infinity, the function approaches 0. Conversely, as x trends towards negative infinity, the function moves towards positive infinity. This tells us about the 'end behavior' of the function without needing to plot its entire graph.“

Rationalizing Expressions

The process of rationalizing expressions involves modifying them to eliminate irrational numbers, or square roots, from the denominators or numerators. This technique simplifies the evaluation of limits, especially when dealing with limits at infinity. This is exactly what we did in Step 3 of our solution. By multiplying the numerator and denominator by the conjugate of the numerator,

\[\left(\sqrt{9x^2 + 1} + 3x\right)\]

we transformed the function into a form that allows easier simplification and limit evaluation. By rationalizing, we could see that the numerator became a constant, and the denominator grew without bounds. Hence, the function approached zero as x increased indefinitely. This approach is often used with functions involving square roots and infinity, as it smoothly paves the way to finding limits.“

Limit Laws

When evaluating limits, especially those involving infinity, limit laws are a set of tools that allow us to break down complex expressions into simpler parts. These laws enable the combination, addition, multiplication, division, and other operations of limits and are fundamental when dissecting complicated functions.

In the given example, we've implicitly used these laws to separate our function into parts that were easier to handle, such as expressing complex square roots in a more manageable way by rationalizing. In addition, knowing that a constant divided by an expression that grows infinitely large yields zero, is part of these vital limit laws. They act like a checklist ensuring that every step to find a limit is grounded on solid mathematical principles.