Q.61 Solution F(y) = Y² - 4y + 3 And G(x) = Sin(x)/x Analysis

Understanding the Problem

This question involves finding the limits of expressions involving two functions, f(y) and g(x). f(y) is a quadratic function, and g(x) is a trigonometric function divided by x. To solve this, we need to:

1. Find the minimum value of f(y).
2. Evaluate the limit of g(x) as x approaches 0.
3. Evaluate the limits of the given expressions using the properties of limits and the results from steps 1 and 2.

Detailed Solution

Let's analyze each part step by step.

Part 1: Finding the Minimum Value of f(y)

The function f(y) = y² - 4y + 3 is a quadratic function. To find its minimum value, we can complete the square or find the vertex of the parabola.

Completing the square:

f(y) = y² - 4y + 3 = (y² - 4y + 4) - 4 + 3 = (y - 2)² - 1

Since (y - 2)² is always non-negative, the minimum value of f(y) occurs when (y - 2)² = 0, which is at y = 2. Thus, the minimum value is:

f(2) = (2 - 2)² - 1 = -1

Alternatively, the vertex of the parabola ay² + by + c is given by y = -b / 2a. In this case, a = 1 and b = -4, so:

y = -(-4) / (2 * 1) = 2

Plugging this back into f(y) gives:

f(2) = 2² - 4(2) + 3 = 4 - 8 + 3 = -1

Thus, the minimum value of f(y) is -1.

Part 2: Evaluating the Limit of g(x) as x → 0

The function g(x) = sin(x) / x is a well-known limit in calculus. As x approaches 0, the limit of g(x) is:

lim{x → 0} sin(x) / x = 1

This limit can be found using L'Hôpital's Rule or by recognizing it as a standard limit.

Part 3: Evaluating the Limits of the Given Expressions

Now, we evaluate each of the given options:

(A) lim_{x → 0} [min(f(y)) · g(x)] = -1

We know that min(f(y)) = -1 and lim{x → 0} g(x) = 1. Therefore:

lim_{x → 0} [min(f(y)) · g(x)] = lim_{x → 0} [-1 · (sin(x) / x)] = -1 · lim_{x → 0} sin(x) / x = -1 · 1 = -1

So, option (A) is correct.

(B) lim_{x → 0} [min(f(y)) · 1/g(x)] = -2

We have min(f(y)) = -1 and g(x) = sin(x) / x. Thus, 1/g(x) = x / sin(x). The limit of 1/g(x) as x approaches 0 is:

lim_{x → 0} 1/g(x) = lim_{x → 0} x / sin(x) = 1 (since lim_{x → 0} sin(x) / x = 1)

Therefore:

lim_{x → 0} [min(f(y)) · 1/g(x)] = lim_{x → 0} [-1 · (x / sin(x))] = -1 · lim_{x → 0} x / sin(x) = -1 · 1 = -1

So, option (B) is incorrect.

(C) lim_{x → [infinity]} cot⁻¹((f(x) · g(x))/x²) = π/2

We have f(x) = x² - 4x + 3 and g(x) = sin(x) / x. So:

(f(x) · g(x)) / x² = ((x² - 4x + 3) · (sin(x) / x)) / x² = (x² - 4x + 3) · sin(x) / x³

As x approaches infinity, we analyze the limit:

lim_{x → [infinity]} (x² - 4x + 3) · sin(x) / x³ = lim_{x → [infinity]} (1 - 4/x + 3/x²) · sin(x) / x

Since -1 ≤ sin(x) ≤ 1, and lim_{x → [infinity]} (1 - 4/x + 3/x²) = 1, we have:

lim_{x → [infinity]} (1 - 4/x + 3/x²) · sin(x) / x = 1 · lim_{x → [infinity]} sin(x) / x = 0

Thus:

lim_{x → [infinity]} cot⁻¹((f(x) · g(x)) / x²) = cot⁻¹(0) = π/2

So, option (C) is correct.

(D) lim_{x → 1} f(x) / |x - 1| = 2

We have f(x) = x² - 4x + 3 = (x - 1)(x - 3). So:

f(x) / |x - 1| = (x - 1)(x - 3) / |x - 1|

We need to consider the limit as x approaches 1 from the left and from the right.

As x → 1⁺, |x - 1| = x - 1, so:

lim_{x → 1⁺} (x - 1)(x - 3) / (x - 1) = lim_{x → 1⁺} (x - 3) = 1 - 3 = -2

As x → 1⁻, |x - 1| = -(x - 1), so:

lim_{x → 1⁻} (x - 1)(x - 3) / -(x - 1) = lim_{x → 1⁻} -(x - 3) = -(1 - 3) = 2

Since the left-hand limit and right-hand limit are not equal, the limit does not exist. Therefore, option (D) is incorrect.

Conclusion

The correct options are (A) and (C).

Final Answer: (A) and (C)

Rewrite for Humans

This problem dives into the world of limits and functions, combining quadratic functions with trigonometric functions. Our mission is to evaluate several limits involving the function f(y) = y² - 4y + 3 and g(x) = sin(x) / x. To tackle this, we first need to understand the behavior of these functions, especially around certain points and as x approaches infinity.

Step 1: Finding the Minimum of the Quadratic Function f(y)

The function f(y) = y² - 4y + 3 is a classic parabola. To find its lowest point, or minimum value, we use a technique called "completing the square." This involves rewriting the function in a form that reveals its vertex—the point where it reaches its minimum. Alternatively, we can find the vertex by using the formula y = -b / 2a for the quadratic ay² + by + c.

Completing the square, we get f(y) = (y - 2)² - 1. From this form, it’s clear that the minimum value occurs when y = 2, making the minimum value f(2) = -1. This means the lowest point on the graph of f(y) is at -1. The minimum value is a critical piece of information for the next steps.

Step 2: Understanding the Limit of g(x) as x Approaches 0

The function g(x) = sin(x) / x is a famous one in calculus. Its behavior as x gets closer to 0 is a fundamental concept. This is a standard limit, often introduced early in calculus courses, and it's equal to 1. In mathematical terms, lim{x → 0} sin(x) / x = 1. This limit is vital because it appears in several options, and knowing it allows us to simplify those options greatly.

This limit can be derived using L'Hôpital’s Rule or Squeeze Theorem but remembering it directly saves a lot of time. The limit approaching zero plays a pivotal role in understanding trigonometric functions and their properties.

Step 3: Evaluating the Limit Expressions

Now comes the fun part: putting everything together and evaluating the given limit expressions. We'll use the minimum value of f(y), the limit of g(x), and the properties of limits to determine which options are correct.

(A) lim_{x → 0} [min(f(y)) · g(x)] = -1

This option combines the minimum value of f(y), which is -1, with the limit of g(x) as x approaches 0, which is 1. The expression simplifies to lim_{x → 0} [-1 · (sin(x) / x)]. As x approaches 0, this becomes -1 · 1 = -1. So, option (A) is indeed correct. It illustrates how knowing basic limits and minimum values can lead to straightforward evaluations. This limit expression is a simple application of fundamental concepts.

(B) lim_{x → 0} [min(f(y)) · 1/g(x)] = -2

Here, we need to consider the reciprocal of g(x), which is 1/g(x) = x / sin(x). The limit of this as x approaches 0 is 1. Thus, the expression becomes lim_{x → 0} [-1 · (x / sin(x))], which evaluates to -1 · 1 = -1, not -2. Therefore, option (B) is incorrect. Miscalculating the reciprocal or its limit can lead to a wrong conclusion. The reciprocal limit is an important part of the analysis.

(C) lim_{x → [infinity]} cot⁻¹((f(x) · g(x)) / x²) = π/2

This is where things get a bit more complex. We’re dealing with the inverse cotangent function and a rational expression involving f(x) and g(x). To solve this, we first rewrite the expression (f(x) · g(x)) / x² as [(x² - 4x + 3) · (sin(x) / x)] / x². Simplifying, we get (1 - 4/x + 3/x²) · sin(x) / x. As x approaches infinity, the term (1 - 4/x + 3/x²) approaches 1, and sin(x) / x approaches 0. Therefore, the expression inside the inverse cotangent approaches 0. The limit of cot⁻¹(0) is π/2, making option (C) correct. This shows how limits at infinity require a different perspective and simplification techniques.

(D) lim_{x → 1} f(x) / |x - 1| = 2

This limit involves an absolute value, which means we need to consider both the left-hand and right-hand limits separately. We rewrite f(x) as (x - 1)(x - 3). The expression becomes (x - 1)(x - 3) / |x - 1|. When x approaches 1 from the right (x → 1⁺), |x - 1| = x - 1, and the limit is -2. When x approaches 1 from the left (x → 1⁻), |x - 1| = -(x - 1), and the limit is 2. Since the left-hand and right-hand limits are different, the limit does not exist, making option (D) incorrect. Limits involving absolute values need careful attention to direction and sign changes.

Final Thoughts

In summary, options (A) and (C) are the correct answers. This problem highlighted the importance of understanding limits, function behavior, and trigonometric identities. It showed how to combine these concepts to evaluate complex expressions and demonstrated the need for careful analysis and step-by-step problem-solving.

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Q.61 Solution f(y) = y² - 4y + 3 and g(x) = sin(x)/x: Step-by-Step Analysis