Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Fundamental Theorem of Calculus

Calculus

8. Definite Integrals

Fundamental Theorem of Calculus

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3:01 minutes

Problem 5.3.108

Textbook Question

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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