How Do You Solve Challenging Limits and Analyze Sequence Monotonicity?

For the first one, you could use the quotient rule and for the second one you could use the product rule.
  • #1
xstetsonx
78
0
Got a couple questions please help!

1. lim n>inf (x^4)/(e^(-2x))

How do you do this one? i know you use L hospital rule but you can never get rid of the denominator. or is there a trick i am missing?

2. Determine whether the sequences are increasing, decreasing, or not monotonic.
1.(cos(x))/(3^x)
2.(x-3)/(x+3)
3.1/(3x+6)

can you explain what to do with this one?
 
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  • #2
xstetsonx said:
Got a couple questions please help!

1. lim n>inf (x^4)/(e^(-2x))

How do you do this one? i know you use L hospital rule but you can never get rid of the denominator. or is there a trick i am missing?
You mean x --> infinity? Why would you use L'Hospital's rule? You have an infinity / 0 form.
2. Determine whether the sequences are increasing, decreasing, or not monotonic.
1.(cos(x))/(3^x)
2.(x-3)/(x+3)
3.1/(3x+6)

can you explain what to do with this one?

You could look at their derivatives.
 

FAQ: How Do You Solve Challenging Limits and Analyze Sequence Monotonicity?

What is an infinite sequence?

An infinite sequence is a list of numbers that continues indefinitely, with a pattern or rule that determines the next term in the sequence. For example, the sequence 2, 4, 6, 8, ... has a pattern of adding 2 to the previous term to get the next term.

What is a series?

A series is the sum of the terms in an infinite sequence. For example, the series 1 + 2 + 3 + 4 + ... is the sum of all positive integers and has no finite sum.

What is the difference between a convergent and divergent series?

A convergent series is one where the sum of all the terms approaches a finite number as you add more terms. In contrast, a divergent series is one where the sum of the terms does not approach a finite number, but instead either grows infinitely large or oscillates between different values.

How do you determine the convergence or divergence of a series?

There are various tests that can be used to determine the convergence or divergence of a series, such as the ratio test, the root test, and the integral test. These tests involve comparing the series to known convergent or divergent series or using mathematical properties to determine the behavior of the series.

What are some real-world applications of infinite sequences and series?

Infinite sequences and series are used in many fields, including physics, engineering, and finance. For example, the concept of limit and convergence is essential in understanding the behavior of physical systems, such as in the study of motion and forces. In finance, infinite series are used to calculate compound interest and to model the growth of investments over time.

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