Help with an exercise

[angryllama]

Hello,
im preparing for an exam and im trying to solve old examples of the exam, where i found some exercises that give me problems...

here are the exercises:

2. Is there a limit of function f in x = 1?

. . . . .\(\displaystyle f(x)\, =\, \begin{cases}\dfrac{1\, -\, \cos(\ln(x^4))}{(2\ln(x))^2}&;\, x\, >\, 1 \\ \dfrac{16x^2\, -\, 20x\, +\, 4}{6x^2\, -\, 9x\, +\, 3}&;\, x\, \leq\, 1 \end{cases}\)

4. (a) With the help of Mathematical induction show: \(\displaystyle \, n\, \in\, \mathbb{N}\)

. . . . .\(\displaystyle \dfrac{1}{1\, \cdot\, 5}\, +\, \dfrac{1}{5\, \cdot\, 9}\, +\, \dfrac{1}{9\, \cdot\, 13}\, +\, ...\, +\, \dfrac{1}{(4n\, -\, 3)\, \cdot\, (4n\, +\, 1)}\, =\, \dfrac{n}{4n\, +\, 1}\)

. ..(b) Show that it is a convergent sequence:

. . . . .\(\displaystyle \displaystyle \sum_{n\, =\, 1}^{\infty}\, \dfrac{1}{(4n\, -\, 3)(4n\, +\, 1)}\)

I would be really grateful if someone could solve so i can see the steps how to approach and all.

 

[Deleted member 4993]

Hello,
im preparing for an exam and im trying to solve old examples of the exam, where i found some exercises that give me problems...

here are the exercises:

2. Is there a limit of function f in x = 1?

. . . . .\(\displaystyle f(x)\, =\, \begin{cases}\dfrac{1\, -\, \cos(\ln(x^4))}{(2\ln(x))^2}&;\, x\, >\, 1 \\ \dfrac{16x^2\, -\, 20x\, +\, 4}{6x^2\, -\, 9x\, +\, 3}&;\, x\, \leq\, 1 \end{cases}\)

4. (a) With the help of Mathematical induction show: \(\displaystyle \, n\, \in\, \mathbb{N}\)

. . . . .\(\displaystyle \dfrac{1}{1\, \cdot\, 5}\, +\, \dfrac{1}{5\, \cdot\, 9}\, +\, \dfrac{1}{9\, \cdot\, 13}\, +\, ...\, +\, \dfrac{1}{(4n\, -\, 3)\, \cdot\, (4n\, +\, 1)}\, =\, \dfrac{n}{4n\, +\, 1}\)

. ..(b) Show that it is a convergent sequence:

. . . . .\(\displaystyle \displaystyle \sum_{n\, =\, 1}^{\infty}\, \dfrac{1}{(4n\, -\, 3)(4n\, +\, 1)}\)

I would be really grateful if someone could solve so i can see the steps how to approach and all.

1) At x= 1

\(\displaystyle \displaystyle{f(x) \ = \ \frac{16x^2 - 20x+4}{6x2-9x+3} \ = \ \frac{4(4x-1)(x-1)}{3(2x-1)(x-1)}}\)... now continue

What are your thoughts?

Please share your work with us ...even if you know it is wrong

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[stapel]

im preparing for an exam and im trying to solve old examples of the exam, where i found some exercises that give me problems. I would be really grateful if someone could solve so i can see the steps how to approach and all.

Since you are needing to learn "how to approach" these exercises in the first place, then the first think you need are lessons. (Working one exercise of each type won't teach you how to do these.) So:

2. Is there a limit of function f in x = 1?

. . . . .\(\displaystyle f(x)\, =\, \begin{cases}\dfrac{1\, -\, \cos(\ln(x^4))}{(2\ln(x))^2}&;\, x\, >\, 1 \\ \dfrac{16x^2\, -\, 20x\, +\, 4}{6x^2\, -\, 9x\, +\, 3}&;\, x\, \leq\, 1 \end{cases}\)

For there to be a limit value (that is, for there to be a value that is "reasonable" for this function to have) at x = 1, then what must be true about the values of the "halves" where they meet? To learn, try here. (By the way, I've assumed that you have learned about limits. If that's wrong, then please say so, so we can provide you with lesson links for that topic, too.)

4. (a) With the help of Mathematical induction show:

\(\displaystyle \, n\, \in\, \mathbb{N}\)

. . . . .\(\displaystyle \dfrac{1}{1\, \cdot\, 5}\, +\, \dfrac{1}{5\, \cdot\, 9}\, +\, \dfrac{1}{9\, \cdot\, 13}\, +\, ...\, +\, \dfrac{1}{(4n\, -\, 3)\, \cdot\, (4n\, +\, 1)}\, =\, \dfrac{n}{4n\, +\, 1}\)

To learn how mathematical induction works, try here After you have studied at least two lessons from the link, please attempt the exercise. You'll start with showing that the left-hand side ("LHS") equals the right-hand side ("RHS") when n = 1. Then you'll write out the formula with n = k. Then you'll use that n = k statement to try to prove the formula for n = k + 1.

. ..(b) Show that it is a convergent sequence:

. . . . .\(\displaystyle \displaystyle \sum_{n\, =\, 1}^{\infty}\, \dfrac{1}{(4n\, -\, 3)(4n\, +\, 1)}\)

What method(s) or algorith(s) have they given you for this sort of exercise? Or are you needing to learn all of them? If the latter, please start here. Either way, please reply (after studying lessons first, if needed) with your chosen method and a clear listing of your efforts so far.

Thank you!