Limits with integrals Evaluate the following limits.

lim ∫₂ˣ eᵗ² dt
𝓍→2 ---------------
𝓍 ― 2

Verified step by step guidance

Step 1: Recognize that the given problem involves a limit with an integral. The numerator is the integral ∫₂ˣ eᵗ² dt, and the denominator is (𝓍 − 2). This is a classic case where L'Hôpital's Rule might be applicable because the limit results in an indeterminate form (0/0) as 𝓍 → 2.

Step 2: Apply L'Hôpital's Rule, which states that if the limit of a quotient results in an indeterminate form, you can differentiate the numerator and denominator separately. Begin by differentiating the numerator with respect to 𝓍. The derivative of ∫₂ˣ eᵗ² dt with respect to 𝓍 is eˣ², using the Fundamental Theorem of Calculus.

Step 3: Differentiate the denominator with respect to 𝓍. The derivative of (𝓍 − 2) with respect to 𝓍 is simply 1.

Step 4: Rewrite the limit after applying L'Hôpital's Rule. The new limit becomes lim 𝓍→2 eˣ² / 1, which simplifies to lim 𝓍→2 eˣ².

Step 5: Evaluate the simplified limit. Substitute 𝓍 = 2 into eˣ² to find the value of the limit. The final result is e²², but the calculation of this value is not required for the steps.

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Key Concepts

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

Limits

In calculus, a limit is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points of discontinuity or infinity. Limits are essential for defining derivatives and integrals, forming the backbone of calculus.

Definite Integrals

A definite integral represents the accumulation of quantities, such as area under a curve, over a specified interval. It is denoted by the integral symbol with upper and lower limits, indicating the range of integration. Evaluating definite integrals often involves the Fundamental Theorem of Calculus, which connects differentiation and integration.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of a quotient of functions yields an indeterminate form, one can take the derivative of the numerator and the derivative of the denominator separately, then re-evaluate the limit. This rule is particularly useful in problems involving limits of integrals.

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(g) ∫ ƒ' (g(𝓍))g' (𝓍) d(𝓍) = ƒ(g(𝓍)) + C .

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Textbook Question

Area functions and the Fundamental Theorem Consider the function

ƒ(t) = { t if ―2 ≤ t < 0

t²/2 if 0 ≤ t ≤ 2

and its graph shown below. Let F(𝓍) = ∫₋₁ˣ ƒ(t) dt and G(𝓍) = ∫₋₂ˣ ƒ(t) dt.

(a) Evaluate F(―2) and F(2).

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Textbook Question