Section 2.3: Computing Limits using Laws

Find the limit if it exists.

1.  (a) \(\displaystyle{\lim_{x \to -1} \, \dfrac{x+1}{x-1}}\)           (b) \(\displaystyle{\lim_{x \to -1} \, \dfrac{2x+2}{x^2-1}}\)    solution

2.   (a) \(\displaystyle{\lim_{x \to 0} \, \dfrac{x^2-4x}{x^3+3x}}\)           (b) \(\displaystyle{\lim_{x \to 2} \frac{x^2-2x}{x^2-x-2} }\)    solution

3.   (a) \(\displaystyle{\lim_{x \to -1} 5x^3-3x^2+7x-11}\)        (b) \(\displaystyle{\lim_{x \to 3} \frac{x^2-2x-3}{x-3} }\)    solution

4.   (a) \(\displaystyle{\lim_{h\to 0} \frac{\frac{1}{5+h}-\frac{1}{5}}{h}}\)          (b) \(\displaystyle{\lim_{h \to 0} \frac{\sqrt{9+h}-3}{h} }\)    solution

5.  \(\displaystyle{\lim_{h \rightarrow 0} \: \frac{\sqrt{25+h}-5}{h}} \)    solution

6.  \(\displaystyle{ \lim_{x \rightarrow 4} \: \frac{\sqrt{x}-2}{x-4}} \)    solution  

7.  \(\displaystyle{\lim_{h \rightarrow 0}\: \frac{(2+h)^2-4}{h}} \)     solution  

8.  \(\displaystyle{\lim_{x \rightarrow 2}\: \frac{\frac{1}{2}-\frac{1}{x}}{2-x}} \)   solution1   solution2

9.   \(\displaystyle{\lim_{x \rightarrow 4}\: \frac{\frac{1}{3}-\frac{1}{\sqrt{x+5}}}{x-4}} \)   solution

10.   \(\displaystyle{ \lim_{x \rightarrow 16}\: \frac{16x-x^2}{4-\sqrt{x}}} \)     solution

11.  \(\displaystyle{\lim_{h \rightarrow 0}\: \frac{\frac{1}{(a+h)^2}-\frac{1}{a^2}}{h}} \)   solution

12. \(\displaystyle{\lim_{x \rightarrow -2}\: \frac{|x+2|}{3x+6}} \)     solution

13. \(\displaystyle{\lim_{x \rightarrow 0}\: x^2\sin \frac{1}{x}} \)   solution

14.   If  \( 2x\le f(x)\le x^4-x^2+2 \)   for all \(x\), find the value of   \(\displaystyle{\lim_{x \rightarrow 1}\: f(x)}. \)  solution

15.  Evaluate the limits or state that they do not exist.  solution

(a)  \( \displaystyle{\lim_{x \to 3^-} \frac{2}{(x-3)^3}} \)  (b) \( \displaystyle{\lim_{x \to 3^+ } \frac{2}{(x-3)^3}} \)  (c) \( \displaystyle{\lim_{x \to 3} \frac{2}{(x-3)^3}} \)