Settling an indeterminate form by one part alone

Practise against it

Each of these has an answer this slip gives. Work them on paper: if you fall into it, the verdict names it and the question comes back in review.

1. 01

   Let $f(x) = 3x^{2} - x^{5}$. For very large positive $x$, which term dominates, and what is the sign of $f(x)$?

   ·Power functions and first sketches·Introductory

2. 02

   Evaluate the limit. $\displaystyle \lim_{x \to \infty} x\sin\!\left(\frac{3}{x}\right)$

   ·L’Hôpital’s rule·Introductory

3. 03

   Evaluate the limit. $\displaystyle \lim_{x \to \infty} \Bigl(\sqrt{9x^{2} + 4x} - 3x\Bigr)$

   ·Limits·Standard

4. 04

   Evaluate $\displaystyle\lim_{x \to 0^{+}} x^{2}\log x$.

   ·L’Hôpital’s rule·Standard

5. 05

   Evaluate the limit. $\displaystyle \lim_{x \to \infty} \left(1 + \frac{3}{x}\right)^{x}$

   ·L’Hôpital’s rule·Standard

6. 06

   Evaluate the limit. $\displaystyle \lim_{x \to -\infty} \Bigl(\sqrt{x^{2} + 5x} - \sqrt{x^{2} - 3x}\Bigr)$

   ·Limits·Challenging