Problem 10 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

\(\lim _{x \rightarrow \infty} \frac{(2 x+1)^{40}(4 x-1)^{5}}{(2 x+3)^{45}}=\) (a) 16 (b) 24 (c) 32 (d) 8

Step-by-Step Solution

Answer

The limit is 32.

Step 1: Analyze the Expression

The expression is \(\lim _{x \rightarrow \infty} \frac{(2 x+1)^{40}(4 x-1)^{5}}{(2 x+3)^{45}}\). When \(x\) approaches infinity, terms like \(2x+1\), \(4x-1\), and \(2x+3\) dominate because they are much larger than the constant terms.

Step 2: Simplify Dominant Terms

We approximate \((2x+1)\), \((4x-1)\), and \((2x+3)\) by \((2x)\), \((4x)\), and \((2x)\), respectively, since at infinity, the constants have negligible effect compared to \(x\). So the expression simplifies to \(\left(\frac{(2x)^{40}(4x)^{5}}{(2x)^{45}}\right)\).

Step 3: Simplify Expression Further

This becomes \(\frac{(2x)^{40} \cdot (4x)^{5}}{(2x)^{45}}\). Notice that \(4x\) can be rewritten as \((2x)^2\), so it becomes \((2x)^{10}\). Plugging it into the expression gives \(\frac{(2x)^{40} \cdot (2x)^{10}}{(2x)^{45}}\).

Step 4: Simplify Exponents

Now simplify the powers: \((2x)^{40+10} = (2x)^{50}\). Dividing, we have \(\frac{(2x)^{50}}{(2x)^{45}} = (2x)^{50-45} = (2x)^5\).

Step 5: Evaluate Limit

Now compute the limit: \((2x)^5 = 2^5 \cdot x^5 = 32x^5\). As \(x \rightarrow \infty\), the factor of \(x^5\) is irrelevant to limit computations, so the limit simplifies to \(2^5 = 32\). Therefore, the answer is (c) 32.

Key Concepts

Infinity Limits

When evaluating limits as a variable approaches infinity, such as in calculus, we look at how the expressions behave when the variable becomes very large. This particular approach allows us to determine the behavior of functions at "infinity." In the context of the given exercise, we're viewing how each term within the expression behaves as we push the variable \(x\) to increasingly larger values.

• In the expression \((2x + 1)^{40}(4x - 1)^5/(2x + 3)^{45}\), when \(x\) approaches infinity, it's crucial to focus on the terms with the variable \(x\), rather than the constants, because these terms grow substantially larger.
• This kind of limit is common in calculus and provides a foundation for understanding asymptotic behavior in functions.

Understanding infinity limits helps us simplify complex expressions by highlighting and isolating the more influential terms that tend to dominate as \(x\) becomes very large.

Dominant Terms

In calculus, particularly for expressions that approach infinity, recognizing dominant terms becomes critical. Dominant terms are those that grow at the fastest rate as the variable tends toward infinity.In our exercise, the terms with the highest powers of \(x\) are the most dominant. The constants in these terms (like \(1\) or \(-1\) in \(2x+1\) and \(4x-1\)) have minimal impact as \(x\) grows large; thus, they are essentially ignored in limit computations.

• The term \((2x+1)\) is approximated by \(2x\).
• Similarly, \((4x-1)\) is approximated by \(4x\).
• The denominator \((2x+3)\) becomes \(2x\).

By simplifying expressions using dominant terms, we address the main variables affecting a limit, allowing us to easily evaluate the expression as \(x\) approaches infinity.

Simplifying Expressions

Simplifying complex mathematical expressions is a key skill in calculus, and it involves reducing terms to make evaluation easier. In the context of limits, it often means reducing the expression to its most significant components as it approaches an extreme value.

• From our exercise, the original expression simplifies by focusing on the dominant terms: \( \frac{(2x)^{40} (4x)^5}{(2x)^{45}} \).
• Notice that through simplification, all the terms containing constants or lesser significance are omitted for convenience.
• The expression further simplifies to \( \frac{(2x)^{40} \cdot (2x)^{10}}{(2x)^{45}} \), which reduces to \( (2x)^5 \).

These simplifications allow us to evaluate limits more intuitively by stripping away unnecessary complexity and focusing our attention on the terms that truly affect the outcome.

Limit Evaluation

Evaluating limits, especially as \(x\) approaches infinity, is a critical component of calculus, providing insight into the end-behavior of functions.After simplifying the expression \( (2x)^{50}/(2x)^{45} \) to \((2x)^5\), approached as \(x\rightarrow\infty\), we need to consider the factored form of the expression in terms of \(x\).

• The expression \((2x)^5 = 2^5 \cdot x^5\) highlights that while the component \(x^5\) tends to infinity, the limit chiefly considers the coefficient \(2^5\).
• This simplifies the limit to \(32\), since the term \(x^5\) becomes irrelevant to the final simplistic evaluation.

Hence, re-evaluating using these steps gives the final answer, \(32\), which demonstrates how limits work as \(x\) tends towards infinity by focusing on the most significant factors.