# Section 5.2 Calculus 2

Computing Limits of Sequences

## 5.2 Computing Limits of Sequences

### 5.2.1 Limits of Sequences and Functions

• If $$f(x)$$ is a function and $$a_n$$ is a sequence such that $$f(n)=a_n$$ for sufficiently large integers $$n$$, then $$\lim_{x\to\infty}f(x)=L$$ implies $$\lim_{n\to\infty}a_n=L$$.
• Therefore all the rules for evaluating $$\lim_{x\to\infty}f(x)$$ extend to evaluating $$\lim_{n\to\infty}a_n$$.
• Example Use factoring to compute $$\lim_{n\to\infty}\frac{4+n}{n^3+1}$$.
• Example Use L’Hopital’s Rule to prove that any sequence defined by the formula $$a_n=\frac{n^2+3}{4-5n^2}$$ converges to $$-\frac{1}{5}$$.
• Example Use the squeeze theorem to compute $$\lim_{n\to\infty}\frac{\sin n}{n}$$.

### 5.2.2 Common Limits

• The following limits are often useful:
• $$\lim_{n\to\infty} x = x$$
• $$\lim_{n\to\infty} \frac{1}{n} = 0$$
• $$\lim_{n\to\infty} \frac{\ln n}{n} = 0$$
• $$\lim_{n\to\infty} \sqrt[n]{p(n)} = 1$$ where $$p(n)$$ is a polynomial
• $$\lim_{n\to\infty} x^n = 0$$, $$|x|<1$$
• $$\lim_{n\to\infty} (1+\frac{x}{n})^n=e^x$$
• $$\lim_{n\to\infty} \frac{x^n}{n!}=0$$
• Example Find $$\lim_{n\to\infty}\frac{\ln(n^3)}{n}$$.
• Example Find $$\lim_{n\to\infty}\frac{3^n+1}{n!}$$.
• Example Find $$\lim_{n\to\infty}(4n)^{1/n}$$.

### 5.2.3 Monotonic and Bounded Sequences

• A sequence $$\<a_n\>_{n=i}^\infty$$ is bounded if there exist real numbers $$A,B$$ such that $$A\leq a_n\leq B$$ for all integers $$n\geq i$$.
• Example Is the sequence $$\<a_n\>_{n=1}^\infty$$ where $$a_n=\frac{n+1}{n}$$ bounded?
• Example Is the sequence $$\<b_n\>_{n=0}^\infty$$ given by $$b_n=\frac{n}{(-3)^n}$$ bounded?
• A sequence is monotonic if it either never increases or never decreases.
• Example Is the sequence $$\<a_n\>_{n=1}^\infty$$ where $$a_n=\frac{n+1}{n}$$ monotonic?
• Example Is the sequence $$\<b_n\>_{n=0}^\infty$$ given by $$b_n=\frac{n}{(-3)^n}$$ monotonic?
• The Monotonic Sequence Theorem states that all bounded monotonic sequences converge.

### Exercises for 5.2

1. Use factoring to compute $$\displaystyle\lim_{n\to\infty}\frac{n-4n^2}{2n^2+7}$$.
2. Use L’Hopital’s Rule to prove that $$\displaystyle\frac{\ln n}{n}\to 0$$.
3. Use the squeeze theorem to compute $$\displaystyle\lim_{n\to\infty}\frac{\cos n}{n\ln n}$$.
4. Find $$\displaystyle\lim_{n\to\infty}\frac{\sin n + 3n^2}{n^2+1}$$.
5. Find $$\displaystyle\lim_{n\to\infty}\frac{\ln(n^n)}{n^2}$$.
6. Find $$\displaystyle\lim_{n\to\infty}(5n^3)^{2/n}$$.
7. Find $$\displaystyle\lim_{n\to\infty}(\frac{1}{\pi})^{3n}$$.
8. Find $$\displaystyle\lim_{n\to\infty}(\frac{1}{2}+\frac{1}{n})^n$$.
9. Find $$\displaystyle\lim_{n\to\infty} \frac{\frac{(n+2)!}{2^n}}{\frac{3n^2n!}{2^{n+1}}}$$.
10. Based on its first few terms, does the sequence $$\<a_n\>_{n=2}^\infty$$ where $$a_n=\frac{2+n^2}{n^2-1}$$ appear bounded? Monotonic? Does it appear to converge?
11. Based on its first few terms, does the sequence $$\<b_n\>_{n=0}^\infty$$ where $$b_n=(-3)^n$$ appear bounded? Monotonic? Does it appear to converge?
12. Based on its first few terms, does the sequence $$\<y_n\>_{n=1}^\infty$$ where $$y_n=(-\frac{1}{2})^n$$ appear bounded? Monotonic? Does it appear to converge?
13. Prove that $$\displaystyle\lim_{n\to\infty} (1+\frac{x}{n})^n=e^x$$ by considering the function version $$\displaystyle L=\lim_{t\to\infty} (1+\frac{x}{t})^t$$ and taking the natural log of both sides of the equality. Use L’Hopital to solve this limit, showing that $$\ln L=x$$ and therefore $$L=e^x$$.
14. Find $$\lim_{n\to\infty}\frac{n!\cos n}{(n+1)!}$$.
15. Find $$\lim_{n\to\infty}\frac{(3+n)^n}{n^n}$$.
16. Which of these statements seems most appropriate for describing the sequence whose initial terms are $$\<\frac{1}{4},-\frac{1}{6},\frac{1}{8},-\frac{1}{10}, \frac{1}{12},\dots\>$$?
• The sequence is bounded and monotonic, so it converges by the Monotonic Sequence Theorem.
• The sequence is not monotonic and not bounded, so it diverges by the Monotonic Sequence Theorem.
• The sequence is bounded, but not monotonic, so the Monotonic Sequence Theorem doesn’t apply. However, it does appear to converge to $$0$$ anyway.

Solutions 1-13

Solutions 14-16

### Textbook References

• University Calculus: Early Transcendentals (3rd Ed)
• 9.1