2.1.1 Calculating limits with limit laws

Suppose $\displaystyle\lim_{x \rightarrow a} f(x)=0$ and $\displaystyle\lim_{x \rightarrow a} g(x)=0$. Which of the following limits can you compute, given this information?

Give two functions $f(x)$ and $g(x)$ that satisfy $\displaystyle\lim_{x \rightarrow 3}f(x)=\displaystyle\lim_{x \rightarrow 3}g(x)=0$ and $\displaystyle\lim_{x \rightarrow 3} \dfrac{f(x)}{g(x)}=10$.

Give two functions $f(x)$ and $g(x)$ that satisfy $\displaystyle\lim_{x \rightarrow 3}f(x)=\displaystyle\lim_{x \rightarrow 3}g(x)=0$ and $\displaystyle\lim_{x \rightarrow 3} \dfrac{f(x)}{g(x)}=0$.

Give two functions $f(x)$ and $g(x)$ that satisfy $\displaystyle\lim_{x \rightarrow 3}f(x)=\displaystyle\lim_{x \rightarrow 3}g(x)=0$ and $\displaystyle\lim_{x \rightarrow 3} \dfrac{f(x)}{g(x)}=\infty$.

Suppose $\displaystyle\lim_{x \rightarrow a}f(x)=\displaystyle\lim_{x \rightarrow a}g(x)=0$. What are the possible values of $\displaystyle\lim_{x \rightarrow a}\dfrac{f(x)}{g(x)}$?

$\displaystyle\lim_{t \rightarrow 10} \dfrac{2(t-10)^2}{t}$

$\displaystyle\lim_{y \rightarrow 0} \dfrac{(y+1)(y+2)(y+3)}{\cos y}$

$\displaystyle\lim_{x \rightarrow 3} \left(\dfrac{4x-2}{x+2}\right)^4$

$\ds \lim_{t\to -3} \left(\frac{1-t}{\cos(t)}\right)$

$\ds \lim_{h \to 0} \frac{(2+h)^2-4}{2h}$

$\ds \lim_{t\to -2} \left(\frac{t-5}{t+4}\right)$

$\ds \lim_{x\to 1} \sqrt{5x^3 + 4}$

$\displaystyle\lim_{t\rightarrow -1} \left(\frac{t-2}{t+3}\right)$

$\lim\limits_{x\rightarrow 1}\dfrac{\log(1+x)-x}{x^2}$

$\displaystyle\lim_{x\rightarrow 2} \left(\frac{x-2}{x^2-4}\right)$

$\ds\lim\limits_{x\rightarrow 4}\dfrac{x^2-4x}{x^2-16}$

$\lim\limits_{x\rightarrow 2}\dfrac{x^2+x-6}{x-2}$

$\ds \lim_{x \to -3} \frac{x^2-9}{x+3}$

$\displaystyle\lim_{t \rightarrow 2} \frac{1}{2}t^4-3t^3+t$

$\ds \lim_{x\to -1} \frac{\sqrt{x^2+8}-3}{x+1}$.

$\displaystyle \lim_{x\rightarrow 1} \frac{\sqrt{x+2}-\sqrt{4-x}}{x-1}$

$\ds \lim_{x\to 3} \frac{\sqrt{x-2}-\sqrt{4-x}}{x-3}$.

$\displaystyle\lim_{x \rightarrow 0}-x^2\cos\left(\frac{3}{x}\right)$

$\lim\limits_{x\rightarrow 0}x\sin^2\left(\dfrac{1}{x}\right)$

$\displaystyle\lim_{w \rightarrow 5} \dfrac{2w^2-50}{(w-5)(w-1)}$

$\displaystyle\lim_{r \rightarrow -5} \dfrac{r}{r^2+10r+25}$

$\displaystyle\lim_{x \rightarrow -1}\sqrt{\dfrac{x^3+x^2+x+1}{3x+3}}$

$\displaystyle\lim_{x \rightarrow 0} \dfrac{x^2+2x+1}{3x^5-5x^3}$

$\displaystyle\lim_{t \rightarrow 7} \dfrac{t^2x^2+2tx+1}{t^2-14t+49}$, where $x$ is a positive constant

$\displaystyle\lim_{d \rightarrow 0} x^5-32x+15$, where $x$ is a constant

$\displaystyle\lim_{x \rightarrow 1} (x-1)^2\sin\left[\left(\dfrac{x^2-3x+2}{x^2-2x+1}\right)^2+15\right]$

Evaluate

or explain why this limit does not exist.

$\lim\limits_{x\rightarrow2}\dfrac{x^2-4}{x^2-2x}$

$\displaystyle\lim_{x \rightarrow 5} \dfrac{(x-5)^2}{x+5}$

Evaluate $\ds\lim_{t \to \frac{1}{2}}\dfrac{\frac{1}{3t^2}+\frac{1}{t^2-1}}{2t-1}$ .

Evaluate $\ds\lim_{x \to 0}\left( 3+\dfrac{|x|}{x}\right)$.

Evaluate $\ds\lim_{d \to -4}\dfrac{|3d+12|}{d+4}$

Evaluate $\ds\lim_{x \to 0}\dfrac{5x-9}{|x|+2}$.

Suppose $\displaystyle\lim_{x \rightarrow -1} f(x)=-1$. Evaluate $\displaystyle\lim_{x \rightarrow -1} \dfrac{xf(x)+3}{2f(x)+1}$.

Find the value of the constant $a$ for which $\lim\limits_{x\rightarrow -2}\dfrac{x^2+ax+3}{x^2+x-2}$ exists.

Suppose $f(x)=2x$ and $g(x)=\frac{1}{x}$. Evaluate the following limits.

$\ds \lim_{x\to 2} \frac{\sqrt{x+7}-\sqrt{11-x}}{2x-4}$.

$\ds \lim_{t\to 1} \frac{3t-3}{2 - \sqrt{5-t}}$.

The curve $y=f(x)$ is shown in the graph below. Sketch the graph of $y=\dfrac{1}{f(x)}$.

The graphs of functions $f(x)$ and $g(x)$ are shown in the graphs below. Use these to sketch the graph of $\dfrac{f(x)}{g(x)}$.

Let $f(x) = \frac{1}{x}$ and $g(x) = \frac{-1}{x}$.

Suppose

Suppose