In Exercises 27-30, use Theorem 4 to determine the limit of the sequence. an=tan^-1(e^-n)

Name: Problem 30, Chapter 10: Calculus for AP
Uploaded: 2019-07-11T06:04:46-08:00
Duration: 45 s
Channel: Fuzail Shakir
Description: In Exercises 27-30, use Theorem 4 to determine the limit of the sequence. an=tan^-1(e^-n)

step 1 All right, so by theorem one, we note that the limit as n approaches infinity of e to the power

In Exercises 27-30, use Theorem 4 to determine the limit of...

Key Concepts

Inverse Trigonometric Function

Inverse trigonometric functions, like the arctangent, are functions that yield an angle corresponding to a given trigonometric ratio. These functions are continuous on their domains and are often used in calculus to transition from algebraic to trigonometric perspectives. Their continuity makes them suitable for applying the continuous function theorem when they are composed with other sequences.

Limit of a Sequence

This concept involves understanding how the values of a sequence behave as the index approaches infinity. A sequence converges if its terms get arbitrarily close to a specific number, known as the limit. Analyzing the limit of a sequence is a fundamental aspect of calculus and real analysis, and many theorems are dedicated to establishing criteria under which sequences converge.

Continuous Function Theorem

The continuous function theorem (or theorem on the limit of composite functions) states that if a function is continuous at a point and a sequence converges to that point, then the composition of the function and the sequence converges to the function evaluated at the limit of the sequence. This theorem is essential when determining the limit of a sequence defined through a continuous transformation of another sequence.

Exponential Decay

Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. In sequences, terms defined using an exponentially decaying function, such as those involving e raised to a negative power, tend to approach zero as the index increases. This characteristic is crucial in limit evaluations where the decaying component drives the behavior of the sequence.

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