Suppose \(\displaystyle\lim_{x \rightarrow a} f(x)=0\) and \(\displaystyle\lim_{x \rightarrow a} g(x)=0\text{.}\) 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 \(\displaystyle\lim_{x \rightarrow 3} \dfrac{f(x)}{g(x)}=10\text{.}\)
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 \(\displaystyle\lim_{x \rightarrow 3} \dfrac{f(x)}{g(x)}=0\text{.}\)
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 \(\displaystyle\lim_{x \rightarrow 3} \dfrac{f(x)}{g(x)}=\infty\text{.}\)
Suppose \(\displaystyle\lim_{x \rightarrow a}f(x)=\displaystyle\lim_{x \rightarrow a}g(x)=0\text{.}\) What are the possible values of \(\displaystyle\lim_{x \rightarrow a}\dfrac{f(x)}{g(x)}\text{?}\)
\(\displaystyle\lim_{t \rightarrow 10} \dfrac{2(t-10)^2}{t}\)
\(\displaystyle\lim_{y \rightarrow 0} \dfrac{(y+1)(y+2)(y+3)}{\cos y}\)
\(\displaystyle\lim_{x \rightarrow 3} \left(\dfrac{4x-2}{x+2}\right)^4\)
\(\ds \lim_{t\to -3} \left(\frac{1-t}{\cos(t)}\right)\)
\(\ds \lim_{h \to 0} \frac{(2+h)^2-4}{2h}\)
\(\ds \lim_{t\to -2} \left(\frac{t-5}{t+4}\right)\)
\(\ds \lim_{x\to 1} \sqrt{5x^3 + 4}\)
\(\displaystyle\lim_{t\rightarrow -1} \left(\frac{t-2}{t+3}\right)\)
\(\lim\limits_{x\rightarrow 1}\dfrac{\log(1+x)-x}{x^2}\)
\(\displaystyle\lim_{x\rightarrow 2} \left(\frac{x-2}{x^2-4}\right)\)
\(\ds\lim\limits_{x\rightarrow 4}\dfrac{x^2-4x}{x^2-16}\)
\(\lim\limits_{x\rightarrow 2}\dfrac{x^2+x-6}{x-2}\)
\(\ds \lim_{x \to -3} \frac{x^2-9}{x+3}\)
\(\displaystyle\lim_{t \rightarrow 2} \frac{1}{2}t^4-3t^3+t\)
\(\ds \lim_{x\to -1} \frac{\sqrt{x^2+8}-3}{x+1}\text{.}\)
\(\ds \lim_{x\to 2} \frac{\sqrt{x+7}-\sqrt{11-x}}{2x-4}\text{.}\)
\(\displaystyle \lim_{x\rightarrow 1} \frac{\sqrt{x+2}-\sqrt{4-x}}{x-1}\)
\(\ds \lim_{x\to 3} \frac{\sqrt{x-2}-\sqrt{4-x}}{x-3}\text{.}\)
\(\ds \lim_{t\to 1} \frac{3t-3}{2 - \sqrt{5-t}}\text{.}\)
\(\displaystyle\lim_{x \rightarrow 0}-x^2\cos\left(\frac{3}{x}\right)\)
\(\displaystyle\lim_{x \rightarrow 0}\dfrac{x^4\sin\left(\frac{1}{x}\right)+5x^2\cos\left(\frac{1}{x}\right)+2}{(x-2)^2}\)
\(\lim\limits_{x\rightarrow 0}x\sin^2\left(\dfrac{1}{x}\right)\)
\(\displaystyle\lim_{w \rightarrow 5} \dfrac{2w^2-50}{(w-5)(w-1)}\)
\(\displaystyle\lim_{r \rightarrow -5} \dfrac{r}{r^2+10r+25}\)
\(\displaystyle\lim_{x \rightarrow -1}\sqrt{\dfrac{x^3+x^2+x+1}{3x+3}}\)
\(\displaystyle\lim_{x \rightarrow 0} \dfrac{x^2+2x+1}{3x^5-5x^3}\)
\(\displaystyle\lim_{t \rightarrow 7} \dfrac{t^2x^2+2tx+1}{t^2-14t+49}\text{,}\) where \(x\) is a positive constant
\(\displaystyle\lim_{d \rightarrow 0} x^5-32x+15\text{,}\) 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 \(\ds \lim_{x\rightarrow 0} x^{1/101} \sin\big(x^{-100}\big)\) 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 \(\ds\lim_{x \to 0}\left( 3+\dfrac{|x|}{x}\right) \text{.}\)
Evaluate \(\ds\lim_{d \to -4}\dfrac{|3d+12|}{d+4}\)
Evaluate \(\ds\lim_{x \to 0}\dfrac{5x-9}{|x|+2}\text{.}\)
Suppose \(\displaystyle\lim_{x \rightarrow -1} f(x)=-1\text{.}\) Evaluate \(\displaystyle\lim_{x \rightarrow -1} \dfrac{xf(x)+3}{2f(x)+1}\text{.}\)
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 \(g(x)=\frac{1}{x}\text{.}\) Evaluate the following limits.
The curve \(y=f(x)\) is shown in the graph below. Sketch the graph of \(y=\dfrac{1}{f(x)}\text{.}\)
The graphs of functions \(f(x)\) and \(g(x)\) are shown in the graphs below. Use these to sketch the graph of \(\dfrac{f(x)}{g(x)}\text{.}\)
Suppose the position of a white ball, at time \(t\text{,}\) is given by \(s(t)\text{,}\) and the position of a red ball is given by \(2s(t)\text{.}\) Using the definition from Section 1.2 of the velocity of a particle, and the limit laws from this section, answer the following question: if the white ball has velocity 5 at time \(t=1\text{,}\) what is the velocity of the red ball?
Let \(f(x) = \frac{1}{x}\) and \(g(x) = \frac{-1}{x}\text{.}\)
Suppose
Suppose