Additional exercises

Exercises · Limits of sequences · with worked solutions · PDF

Exercise 1

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \frac{2^{-n} + 3\: n^3}{\log^6 n +2 - n^5} \]

Exercise 2

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \frac{\log n + 3\: n^3 \: \log n}{2^{1/n} + n^5} \]

Exercise 3

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \frac{n^2 + \log^3 n}{n \: \log n +1 } \]

Exercise 4

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \frac{n^{2+ \frac{1}{n}}}{1 + n^{3-\frac{2}{n}}} \]

Exercise 5

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \left( \frac{n}{n+1}\right)^{2 \:n} \]

Exercise 6

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \left( \frac{n+2}{n+1}\right)^{n^2} \]

Exercise 7

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \left( 1 + \frac{3}{n}\right)^{-n} \]

Exercise 8

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \left( 1 - \frac{2}{n}\right)^{n} \]

Exercise 9

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \frac{n + \log (n^2) - 2^n}{(\log n)^3 + n^2} \]

Exercise 10

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \frac{8 + 2\:n -n ^{4/3}}{n^{-1/3} + 3\:n - n ^{5/3}} \]

Exercise 11

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} n \left( \sqrt{n^2+3} - n \right) \]

Exercise 12

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \frac{n^2 + \log(5\: n^3) + (\log n)^2}{2\: (\log n)^3 + n\: \log n} \]

Exercise 13

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \frac{8 + 2\: n^2 - n^{4/3}}{n^{-1/3} + 3\: n^2 - n^{5/3}} \]

Exercise 14

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \left( \sqrt{n^4+1} - n\: \sqrt{n^2+1} \right) \]

Exercise 15

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \frac{n^2 + 3^n + 2^{-n}}{2^{2\:n} +(\log n)^3} \]

Exercise 16

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \frac{1}{n} \: (\sin^2 n) \cdot \sin (n^2) \]

Exercise 17

Compute, if it exists, the limit of the sequence, justifying the steps taken:

\[ \lim_{n \rr \ip} \frac{\cos (\pi \: n)}{\pi \: n} + \log \left( \sin \frac{\pi}{n}\right) \]