Evaluate

lim [ ∫₂ˣ √(t² + t + 3dt) ] / (𝓍² ―4)
𝓍→2

Verified step by step guidance

Step 1: Recognize that the problem involves evaluating a limit as x approaches 2. The numerator is an integral expression ∫₂ˣ √(t² + t + 3) dt, and the denominator is (x² - 4). This suggests the use of L'Hôpital's Rule since the limit may result in an indeterminate form (0/0).

Step 2: Begin by analyzing the numerator. The integral ∫₂ˣ √(t² + t + 3) dt represents the area under the curve √(t² + t + 3) from t = 2 to t = x. To differentiate this with respect to x, apply the Fundamental Theorem of Calculus, which states that the derivative of an integral with a variable upper limit is the integrand evaluated at that upper limit. Thus, the derivative of the numerator is √(x² + x + 3).

Step 3: Differentiate the denominator, x² - 4, with respect to x. The derivative is 2x.

Step 4: Apply L'Hôpital's Rule, which states that if the limit results in an indeterminate form (0/0 or ∞/∞), you can take the derivative of the numerator and the derivative of the denominator separately. The new limit becomes lim (x → 2) [√(x² + x + 3)] / [2x].

Step 5: Substitute x = 2 into the simplified expression to evaluate the limit. This involves plugging x = 2 into √(x² + x + 3) and 2x. Simplify the resulting expression to find the final value of the limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this question, we are interested in the limit of a ratio as x approaches 2, which requires evaluating how both the numerator and denominator behave near this point.

Definite Integrals

A definite integral represents the accumulation of quantities, such as area under a curve, over a specified interval. In this case, the integral from 2 to x of the function √(t² + t + 3) is crucial for determining the value of the numerator in the limit expression.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. If the limit results in such a form, this rule allows us to differentiate the numerator and denominator separately to find the limit, which is applicable in this problem as both the integral and the denominator approach 0 as x approaches 2.

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