Find lim n → ∞ x n if x n = n n + 1 + n

Given x n = n n + 1 + n = 1 n + 1 n + 1 = 1 1 + 1 n + 1 Now, n → ∞ ⇒ 1 n → 0 , So, lim n → ∞ x n = 1 1 + 0 + 1 = 1 2

Find lim n → ∞ x n if x n = n 2 + 4 n n 3 - 3 n 2 3

Given x n = n 2 + 4 n n 3 - 3 n 2 3 = n 2 1 + 4 n n 3 1 - 3 n 3 = n · 1 + 4 n n · 1 - 3 n 3 As n → ∞ , n = n & 1 n → 0 So, lim n → ∞ x n = 1 + 0 1 - 0 3 = 1

Find lim n → ∞ x n if x n = 1 2 n cos n 3 - 3 n 6 n + 1

We have x n = cos n 3 2 n - 3 n 6 n + 1 = cos n 3 2 n - 3 6 + 1 n As n → ∞ , 1 n → 0 & cos n 3 becomes an oscillating number between - 1 & 1 So, lim n → ∞ cos n 3 2 n - 3 6 + 1 n = 0 - 3 6 = - 1 2

Find lim n → ∞ x n if x n = 1 + 1 2 + 1 4 + … . + 1 2 n 1 + 1 3 + 1 9 + … + 1 3 n

We have x n = 1 + 1 2 + 1 4 + … . + 1 2 n 1 + 1 3 + 1 9 + … + 1 3 n The numerator is a GP with first term a 1 = 1 , common ratio r 1 = 1 2 , and number of terms n ; and denominator is another GP with first term a 2 = 1 , common ratio r 2 = 1 3 and number of terms n . As n → ∞ , both become infinite GPs. Using a + a r + a r 2 + . . . . = a 1 - r , r < 1 , we get lim n → ∞ x n = 1 1 - 1 2 1 1 - 1 3 = 4 3

E M B I B E

PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Evaluation of Limits of Sequences

I A Maron Solutions for Chapter: Introduction to Mathematical Analysis, Exercise 7: Evaluation of Limits of Sequences

Author:I A Maron

I A Maron Mathematics Solutions for Exercise - I A Maron Solutions for Chapter: Introduction to Mathematical Analysis, Exercise 7: Evaluation of Limits of Sequences

Attempt the practice questions on Chapter 1: Introduction to Mathematical Analysis, Exercise 7: Evaluation of Limits of Sequences with hints and solutions to strengthen your understanding. PROBLEMS IN CALCULUS OF ONE VARIABLE solutions are prepared by Experienced Embibe Experts.

Questions from I A Maron Solutions for Chapter: Introduction to Mathematical Analysis, Exercise 7: Evaluation of Limits of Sequences with Hints & Solutions

HARD

Mathematics

IMPORTANT

PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Evaluation of Limits of Sequences>Q 1.7.4.

Find $\lim_{n \to \infty} x_{n}$ if $x_{n} = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{n (n + 1)}$

MEDIUM

Mathematics

IMPORTANT

PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Evaluation of Limits of Sequences>Q 1.7.5.

Find $\lim_{n \to \infty} x_{n}$ if $x_{n} = \frac{\sqrt{n}}{\sqrt{n + 1} + \sqrt{n}}$

MEDIUM

Mathematics

IMPORTANT

PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Evaluation of Limits of Sequences>Q 1.7.5.

Find $\lim_{n \to \infty} x_{n}$ if $x_{n} = \frac{\sqrt{n^{2} + 4 n}}{\sqrt[3]{n^{3} - 3 n^{2}}}$

HARD

Mathematics

IMPORTANT

PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Evaluation of Limits of Sequences>Q 1.7.5.

Find $\lim_{n \to \infty} x_{n}$ if $x_{n} = \sqrt[3]{1 - n^{3}} + n$

HARD

Mathematics

IMPORTANT

PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Evaluation of Limits of Sequences>Q 1.7.5.

Find $\lim_{n \to \infty} x_{n}$ if $x_{n} = \frac{1}{2 n} \cos n^{3} - \frac{3 n}{6 n + 1}$

HARD

Mathematics

IMPORTANT

PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Evaluation of Limits of Sequences>Q 1.7.5.

Find $\lim_{n \to \infty} x_{n}$ if $x_{n} = \frac{2 n}{2 n^{2} - 1} \cos \frac{n + 1}{2 n - 1} - \frac{n}{1 - 2 n} \frac{n {(- 1)}^{n}}{n^{2} + 1}$

HARD

Mathematics

IMPORTANT

PROBLEMS IN CALCULUS OF ONE VARIABLE>Chapter 1 - Introduction to Mathematical Analysis>Evaluation of Limits of Sequences>Q 1.7.5.

Find $\lim_{n \to \infty} x_{n}$ if $x_{n} = \frac{1 + \frac{1}{2} + \frac{1}{4} + \dots . + \frac{1}{2^{n}}}{1 + \frac{1}{3} + \frac{1}{9} + \dots + \frac{1}{3^{n}}}$

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