Limits

Use interval notation to fill in the blanks that follow. Your

answers will involve$\delta ,\epsilon , N, \mathrm{and}/\mathrm{or} M$.

If $\underset{x\to 1^{+}}{\lim }f\left(x\right)=\infty$ then for all $M>0$ there is some $\delta >0$ such that if $x\in \_\_\_\_\_ \mathrm{then} \in \_\_\_\_\_$.

Sketch a labeled graph that illustrates what is going on in the proof of Theorem 1.6 in the reading. Your graph should include two different $ϵ$-bars and a graphical reason that they cannot overlap.

Sketch a labeled graph that illustrates what is going on in the proof of Theorem 1.8 in the reading. Your graph should include two different δ-bars and a graphical reason why they combine to make a punctured delta interval 

Suppose f is a function with f(2) = 5 where for all $ϵ$ > 0, there is some δ > 0 such that if x ∈ (2 − δ, 2) ∪ (2, 2 + δ), then f(x) ∈ (3 − $ϵ$, 3 + $ϵ$). Sketch a possible graph of f.

If $x\in (1.5, 2.5)$, what is the largest interval $I=\left(4-\epsilon ,4+\epsilon \right)$ for which we can guarantee that $x^2\in I$ ?

It is false that $\underset{x\to 1}{\lim }\left(x+1.01\right)=2$. Express this fact in a mathematical sentence involving $\delta$ and $\epsilon$, to show how the formal definition of limit fails in this case.

For each limit  $\underset{x\to c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\delta$such that if $x\in (c-\delta ,c)\cup (c,c+\delta ) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in (\mathrm{L}-\epsilon ,\mathrm{L}+\epsilon )\mathit{ }$.

$\underset{\mathrm{x}\to 2}{\lim }{\mathrm{x}}^3=8, \mathrm{\epsilon }=0.5$

For each limit  $\underset{x\to c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\delta$such that if $x\in (c-\delta ,c)\cup (c,c+\delta ) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in (\mathrm{L}-\epsilon ,\mathrm{L}+\epsilon )\mathit{ }$.

$\underset{\mathrm{x}\to 2}{\lim }{\mathrm{x}}^3=8, \mathrm{\epsilon }=0.25$

For each limit $\underset{x\to c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\delta$such that if $x\in (c-\delta ,c)\cup (c,c+\delta ) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in (\mathrm{L}-\epsilon ,\mathrm{L}+\epsilon )\mathit{ }$.

$\underset{\mathrm{x}\to 5}{\lim }\sqrt{\mathrm{x}-1}=2, \mathrm{\epsilon }=1$

For each limit $\underset{\mathrm{x}\to \mathrm{c}}{\lim }\mathrm{f}\left(\mathrm{x}\right)=\mathrm{L}$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\delta$such that if $x\in (c-\delta ,c)\cup (c,c+\delta ) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in (\mathrm{L}-\epsilon ,\mathrm{L}+\epsilon )\mathit{ }a$.

$\underset{\mathrm{x}\to 5}{\lim }\sqrt{\mathrm{x}-1}=2, \mathrm{\epsilon }=0.2$

For each limit $\underset{x\to c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\delta$such that if $x\in (c-\delta ,c)\cup (c,c+\delta ) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in (\mathrm{L}-\epsilon ,\mathrm{L}+\epsilon )\mathit{ }$.

$\underset{\mathrm{x}\to \mathrm{\pi }}{\lim }\sin \left(\mathrm{x}\right)=0, \mathrm{\epsilon }=\frac{\sqrt{2}}{2}$

For each limit $\underset{x\to c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\delta$ such that if $x\in (c-\delta ,c)\cup (c,c+\delta ) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in (\mathrm{L}-\epsilon ,\mathrm{L}+\epsilon )\mathit{ }$.

$\underset{\mathrm{x}\to 0}{\lim }\frac{1-\cos \left(\mathrm{x}\right)}{\mathrm{x}}=0, \mathrm{\epsilon }=\frac{1}{2}$

For each limit $\underset{x\to c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\delta$such that if $x\in (c-\delta ,c)\cup (c,c+\delta ) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in (\mathrm{L}-\epsilon ,\mathrm{L}+\epsilon )\mathit{ }$.

$\underset{\mathrm{x}\to 0}{\lim }\frac{1-\cos \left(\mathrm{x}\right)}{\mathrm{x}}=0, \mathrm{\epsilon }=\frac{1}{4}$

For each limit $\underset{x\to c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\delta$ such that if $x\in (c-\delta ,c)\cup (c,c+\delta ) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in (\mathrm{L}-\epsilon ,\mathrm{L}+\epsilon )\mathit{ }$.

$\underset{\mathrm{x}\to 3}{\lim }\left(2-{\mathrm{x}}^2\right)=-7, \mathrm{\epsilon }=0.01$

For each limit $\underset{x\to c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\delta$such that if $x\in (c-\delta ,c)\cup (c,c+\delta ) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in (\mathrm{L}-\epsilon ,\mathrm{L}+\epsilon )\mathit{ }$.

$\underset{\mathrm{x}\to 3}{\lim }\left(2-{\mathrm{x}}^2\right)=-7, \mathrm{\epsilon }=0.001$

It is false that $\underset{x\to \infty }{\lim }\frac{1000}{x}=\infty$. Express this fact in a mathematical sentence involving $M \mathrm{and} N$, to show how the formal definition of limit fails in this case. 

Show that the limit as $x\to 2 \mathrm{of} \mathrm{f}\left(\mathrm{x}\right)=\sqrt{\mathrm{x}-1.1}$ is not equal to 1, by finding an  $\epsilon >0$ for which there is no corresponding $\delta >0$ satisfying the formal definition of limit.

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to -3}{\lim }\sqrt{x+7}=2$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to 2}{\lim }\frac{x^2-4}{x+2}=2$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to -1}{\lim }\left(x^3-2\right)=-3$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to 1^{-}}{\lim }\sqrt{1-x}=0$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$v and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to 2}{\lim }\frac{x^2-4}{x-2}=4$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to 1}{\lim }\left(x^2-3\right)=-2$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to 0^{+}}{\lim }\sqrt{x}=0$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to 3^{-}}{\lim }\left(4-x^2\right)=-5$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to -2^{+}}{\lim }\frac{1}{x+2}=\infty$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to 0^{-}}{\lim }\frac{1}{x}=-\infty$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to 2^{+}}{\lim }\frac{1}{2^x-4}=\infty$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to 0^{+}}{\lim }\frac{1}{1-e^x}=-\infty$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to \infty }{\lim }\frac{x}{1-2x}=-5$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to -\infty }{\lim }\frac{1}{x^2+1}=0$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

 $\underset{x\to \infty }{\lim }\left(x^3+x+1\right)=\infty$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to -\infty }{\lim }\left(1-3x\right)=\infty$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{h\to 0}{\lim }\frac{{\left(2+h\right)}^2-4}{h}=4$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{h\to 0}{\lim }\frac{{\displaystyle \frac{1}{2+h}}-{\displaystyle \frac{1}{2}}}{h}=-\frac{1}{4}$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph. 

$\underset{x\to c^{+}}{\lim }f\left(x\right)=-\infty$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph. 

$\underset{x\to c^{-}}{\lim }f\left(x\right)=\infty$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph.

$\underset{x\to -\infty }{\lim }f\left(x\right)=L$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph. 

$\underset{x\to -\infty }{\lim }f\left(x\right)=-\infty$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph. 

$\underset{x\to -\infty }{\lim }f\left(x\right)=\infty$

Write each limit in Exercises 19–42 as a formal statement involving $\delta ,\epsilon ,N, \mathrm{and}/\mathrm{or} M,$ and sketch a graph that illustrates the roles of these constants. In the last six exercises you may take function to be any appropriate graph. 

$\underset{x\to \infty }{\lim }f\left(x\right)=-\infty$

For each limit $\underset{x\to c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\delta$ such that if $x\in (c-\delta ,c)\cup (c,c+\delta ) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in (\mathrm{L}-\epsilon ,\mathrm{L}+\epsilon )\mathit{ }$

$\underset{x\to -1}{lim}\frac{x^2-2x-3}{x+1}=-4,\epsilon =1$

For each limit $\underset{x\to c}{lim}f\left(x\right)=L$in Exercises 43–54, use graphs and algebra to approximate the largest value of $\delta$ such that if $x\in (c-\delta ,c)\cup (c,c+\delta ) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in (\mathrm{L}-\epsilon ,\mathrm{L}+\epsilon )\mathit{ }$

$\underset{x\to -1}{lim}\frac{x^2-2x-3}{x+1}=-4,\epsilon =0.1$

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.

$\underset{x\to 1^{+}}{lim}\frac{1}{x^2-1}=\infty , M=1000, \mathrm{find} \mathrm{largest} \delta >0$

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition. 

$\underset{x\to 1^{+}}{lim}\frac{1}{x^2-1}=\infty , M=10000, \mathrm{find} \mathrm{largest} \delta >0$

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.  

$\underset{x\to \infty }{lim}\frac{3x}{x+1}=3, \epsilon =0.5, \mathrm{find} \mathrm{smallest} N>0$

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N  that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.   

$\underset{x\to \infty }{lim}\frac{3x}{x+1}=3,\epsilon =0.1, \mathrm{find} smallest N>0$

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N  that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.    

$\underset{x\to \infty }{lim}\ln x=\infty , M=100, \mathrm{find} \mathrm{smallest} N>0$

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N  that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition. 

$\underset{x\to \infty }{lim}\ln x=\infty , M=100000, \mathrm{find} \mathrm{smallest} N>0$

For each limit in Exercises 55–64, use graphs and algebra to approximate the largest delta or smallest-magnitude N  that corresponds to the given value of epsilon or M, according to the appropriate formal limit definition.  

$\underset{x\to -\infty }{lim}{3}^x=0, \epsilon =\frac{1}{2}, \mathrm{find} \mathrm{smallest}-\mathrm{magnitude} N<0$