# BC Calculus: Extra Limits Skills Practice. §2.2 and §2.3

• Class: 5H: BC Calculus
• Author: Peter Atlas
• Text: Calculus Finney, Demana, Waits, Kennedy

Evaluate the following limits:

1. $$\displaystyle\lim_{x \to 2}\frac{x^2 - 4}{x^2 + 4} =$$
2. Solution = 0. Substitute in and evaluate to get 0/8
3. $$\displaystyle\lim_{x \to \infty }\frac{4 - x^2}{x^2 - 1} =$$
4. Solution = -1 (Ratio of coefficients of dominant terms.)
5. $$\displaystyle\lim_{x \to 3}\frac{x - 3}{x^2 - 2x - 3} =$$
6. Solution = $$\frac{1}{4}$$ (Factor denominator, reduce, substitue.)
7. $$\displaystyle\lim_{x \to 0 }\frac{|x|}{x} =$$
8. Solution D. N. E. (This is $$y = 1$$ to the right of the y-axis, and $$y = -1$$ to the left of the y-axis. Left- and right-handed limits disagree.)
9. $$\displaystyle\lim_{x \to 0 } \frac {\sin{(3x)}}{\sin{(4x)}} =$$
10. Solution = $$\frac{3}{4}$$. (Using $$\displaystyle\lim_{x \to 0} \frac{\sin{x}}{x} = 1$$ and substitution to get $$\displaystyle \lim_{x \to 0} \frac{\sin{(ax)}}{bx} = \frac{a}{b}$$.)
11. $$\displaystyle\lim_{x \to \infty } \sin {\left( \frac{1}{x} \right) } =$$
12. Solution = 0. (Note this is $$x \to \infty$$, not 0.)
13. $$\displaystyle\lim_{x \to 0 }\sin{\left( \frac{1}{x} \right) } =$$
14. Solution D. N. E. (Oscillates infinitely.)
15. $$\displaystyle\lim_{x \to 25}\frac{\sqrt{x} - 5}{x - 25} =$$
16. Solution = $$\frac{1}{\sqrt{25} + 5}$$. (Rationalize numerator -- or factor denominator -- then reduce. No need to simplify the denominator in the final answer.)
17. $$\displaystyle\lim_{x \to \infty }\frac{3x^2 + 27}{x^3 - 27} =$$
18. Solution = 0. (Dominant term is in denominator.)
19. $$\displaystyle\lim_{x \to 0} x \csc{x} =$$
20. Solution = 1. (This is $$\frac{x}{\sin{x}}$$, which just approaches the reciprocal of 1.)
21. $$\displaystyle\lim_{x \to 0 }|x| =$$
22. Solution = 0. (An uninteresting question, until you get to the next chapter.)