Limits at Infinity: lim[x->-infty](1-4*cbrt[x^2])/(9+10x) = ... = 0 ?

[doughishere]

Im not sure if im pulling this out of my a** or what...im not confident in this answer. Is this right?

Problem 11: Find the limit, as \(\displaystyle x\, \rightarrow\, -\infty,\) of the function \(\displaystyle f(x)\, =\, \dfrac{1\, -\, 4\, \sqrt[3]{\strut x^2\,}}{9\, +\, 10x}\)

Solution:

\(\displaystyle \displaystyle \mbox{(a) }\, \lim_{x \rightarrow -\infty}\, \dfrac{1\, -\, 4\, \sqrt[3]{\strut x^2}}{9\, +\, 10}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, \lim_{x \rightarrow -\infty}\, \dfrac{x\, \left(\frac{1}{x}\, -\, \frac{4}{x^{1/3}}\right)}{x\left(\frac{9}{x}\, +\, 10\right)}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, \lim_{x \rightarrow -\infty}\,\dfrac{\left(\frac{1}{x}\, -\, \frac{4}{x^{1/3}}\right)}{\left(\frac{9}{x}\, +\, 10\right)}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, \dfrac{0\, -\, (0)}{0\, +\, 10}\, =\, \dfrac{0}{10}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, 0\)

 

[Deleted member 4993]

Im not sure if im pulling this out of my a** or what...im not confident in this answer. Is this right?

Problem 11: Find the limit, as \(\displaystyle x\, \rightarrow\, -\infty,\) of the function \(\displaystyle f(x)\, =\, \dfrac{1\, -\, 4\, \sqrt[3]{\strut x^2\,}}{9\, +\, 10x}\)

Solution:

\(\displaystyle \displaystyle \mbox{(a) }\, \lim_{x \rightarrow -\infty}\, \dfrac{1\, -\, 4\, \sqrt[3]{\strut x^2}}{9\, +\, 10}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, \lim_{x \rightarrow -\infty}\, \dfrac{x\, \left(\frac{1}{x}\, -\, \frac{4}{x^{1/3}}\right)}{x\left(\frac{9}{x}\, +\, 10\right)}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, \lim_{x \rightarrow -\infty}\,\dfrac{\left(\frac{1}{x}\, -\, \frac{4}{x^{1/3}}\right)}{\left(\frac{9}{x}\, +\, 10\right)}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, \dfrac{0\, -\, (0)}{0\, +\, 10}\, =\, \dfrac{0}{10}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, 0\)

As far as I can tell, it is correct.

 

[Steven G]

Im not sure if im pulling this out of my a** or what...im not confident in this answer. Is this right?

Problem 11: Find the limit, as \(\displaystyle x\, \rightarrow\, -\infty,\) of the function \(\displaystyle f(x)\, =\, \dfrac{1\, -\, 4\, \sqrt[3]{\strut x^2\,}}{9\, +\, 10x}\)

Solution:

\(\displaystyle \displaystyle \mbox{(a) }\, \lim_{x \rightarrow -\infty}\, \dfrac{1\, -\, 4\, \sqrt[3]{\strut x^2}}{9\, +\, 10}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, \lim_{x \rightarrow -\infty}\, \dfrac{x\, \left(\frac{1}{x}\, -\, \frac{4}{x^{1/3}}\right)}{x\left(\frac{9}{x}\, +\, 10\right)}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, \lim_{x \rightarrow -\infty}\,\dfrac{\left(\frac{1}{x}\, -\, \frac{4}{x^{1/3}}\right)}{\left(\frac{9}{x}\, +\, 10\right)}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, \dfrac{0\, -\, (0)}{0\, +\, 10}\, =\, \dfrac{0}{10}\)

. . . . . . . . . .\(\displaystyle \displaystyle =\, 0\)

Yes, it is correct. Since the power of x is larger in the denominator, this the limit will be 0.

You are most welcome to show us your work but with these limit problems you can verify your answer. In this case you can for example plug in -100,000 for x and see that you get close to 0. Good job.

 

[doughishere]

Thanks both. Follow up...would you consider that "complete?"

When I went about solving this problem I actually substituted values for x to check but I didnt put those down on that attachment.

 

[Steven G]

Thanks both. Follow up...would you consider that "complete?"

When I went about solving this problem I actually substituted values for x to check but I didnt put those down on that attachment.

Yes, what you showed is fine except for the typo where you wrote x*(1/x) where it should say 1*(1/x)

 

[doughishere]

Yes, what you showed is fine except for the typo where you wrote x*(1/x) where it should say 1*(1/x)

Thanks.


Im doing a big latex pdf with all of my calculus answers....its a hugely intensive process getting a page layout customized for each chapter...then working the problems...then entering all the information...then on top of that i print out the pdf and work the problems a second time looking for such errors and things....blah. i endjoy it sometimes its just long.