Problem 99 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

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

Step-by-Step Solution

Answer

In summary, after simplifying the given expression as \(x\) approaches infinity, we find that the limit is equal to 1: \(\lim_{x \rightarrow \infty}\frac{\sqrt{x^{2}+2}}{x-2} = 1\).

Step 1: Identify the expression

We have the expression: \(\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+2}}{x-2}\). Our goal is to find the limit of this expression as x approaches infinity.

Step 2: Simplify the expression

We can simplify the expression by factoring out an \(x^2\) term from the square root in the numerator. This gives us: \(\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}(1+\frac{2}{x^2})}}{x-2}\)

Step 3: Factor x from the numerator

Now, we can factor the x from the square root in the numerator: \(\lim_{x \rightarrow \infty} \frac{x\sqrt{1+\frac{2}{x^2}}}{x-2}\)

Step 4: Cancel out x term

Notice that we can cancel out the x term in both the numerator and the denominator: \(\lim_{x \rightarrow \infty} \frac{\sqrt{1+\frac{2}{x^2}}}{1-\frac{2}{x}}\)

Step 5: Evaluate the limit

Now, as x approaches infinity, the terms \(\frac{2}{x^2}\) and \(\frac{2}{x}\) will approach 0: \(\lim_{x \rightarrow \infty} \frac{\sqrt{1+0}}{1-0}\)

Step 6: Simplify and find the answer

Finally, we can simplify the expression: \(\lim_{x \rightarrow \infty} \frac{\sqrt{1}}{1} = \frac{1}{1} = 1\) The answer is 1.

Key Concepts

Infinity Limit

When we talk about an infinity limit, we are exploring what happens to a function as the variable (often noted as \(x\)) grows without bound. In mathematical terms, this means as \(x\) approaches infinity. In our exercise, \(x\) is approaching infinity within the expression:

• \(\lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+2}}{x-2}\).

What must be determined is how the values of the numerator and denominator behave as \(x\) becomes infinitely large. This concept helps us grasp the idea of function end-behavior—a crucial part of calculus.
Infinity limits allow us to predict and simplify complex behaviors by focusing on dominant terms. Here, both the numerator and the denominator are influenced by terms involving \(x\). Eradicating the lesser influential terms (like constants, or terms decreasing as \(x\) increases) simplifies the computation, making it manageable even when dealing with infinity.

Algebraic Simplification

Algebraic simplification is an essential tool for breaking down mathematical expressions into simpler parts. In the context of limits, simplification helps us understand the behavior of functions. Let’s dissect our exercise:

• Initially, we have \(\frac{\sqrt{x^{2}+2}}{x-2}\).

To simplify this, we factor \(x^2\) out of the square root, rewriting the numerator as \(x\sqrt{1 + \frac{2}{x^2}}\). With this, the expression becomes:

• \(\frac{x\sqrt{1 + \frac{2}{x^2}}}{x-2}\).

By factoring, terms can be cancelled, greatly simplifying the expression. In this case, the \(x\) in the numerator cancels with the \(x\) in the denominator's dominant term, allowing us to move from a more complex expression to a form that’s easier to evaluate:

• \(\frac{\sqrt{1 + \frac{2}{x^2}}}{1 - \frac{2}{x}}\).

This simplification is vital in evaluating the limit by focusing on terms that significantly affect the outcome as \(x\) approaches infinity.

Square Root Simplification

Simplifying square roots can be tricky, but it is necessary for managing expressions effectively. Consider the square root \(\sqrt{x^{2}+2}\) from the given expression. The primary goal is to extract the simplest form without changing its value under the limit.
To achieve square root simplification, notice that \(\sqrt{x^{2}+2}\) can be rewritten by factoring out \(x^{2}\):

• \(x\sqrt{1+\frac{2}{x^2}}\).

This manipulation enables us to separate the \(x\), a critical step that allows further simplification and eventual cancellation of terms between the numerator and denominator. With this simplification, as \(x\) approaches infinity:

• \(\frac{1 + \frac{2}{x^2}}\) approaches \(1\).

Understanding how square root manipulation aids in broader arithmetic processes provides a clearer path to resolving limits, ensuring we comprehend how each component interacts within the problem context.