Help on a calculus question of evaluating the limits

[theking]

evaluate

 

[theking]

im really confused about how to start this problem. Thx so much for helping me

 

[Deleted member 4993]

evaluate

Have you learned L'Hospital's method?

 

[theking]

View attachment 14317
Have you learned L'Hospital's method?

Not yet. My teacher assigned this problem after we finished the limit

 

[Steven G]

If you haven't you learned L'Hospital's method, then use your knowledge of how the difference of cubes factors.

 

[theking]

If you haven't you learned L'Hospital's method, then use your knowledge of how the difference of cubes factors.

The only thing i can think of right now it's i can let the cube root be a letter. but I don't know what to do next

 

[tkhunny]

While thinking about factoring a difference fo cubes, you may wish to think that you already have the two-term factor. Build the related three-term factor.

 

[Steven G]

What do you multiply a-b by to get a^3 - b^3

 

[pka]

The only thing i can think of right now it's i can let the cube root be a letter. but I don't know what to do next

\(\displaystyle \left( {\frac{{\sqrt[3]{{1 + {\pi ^2}x}}-1}}{x}} \right)\left( {\frac{{{{(\sqrt[3]{{1 + {\pi ^2}x}})}^2} - (\sqrt[3]{{1 + {\pi ^2}x}}) + 1}}{{\sqrt[3]{{1 + {\pi ^2}x}}{)^2} - (\sqrt[3]{{1 + {\pi ^2}x}}) + 1}}} \right) = \frac{{1 + {\pi ^2}x - 1}}{{x\left( {\sqrt[3]{{1 + {\pi ^2}x}}{)^2} - (\sqrt[3]{{1 + {\pi ^2}x}}) + 1} \right)}}\)

 

[theking]

\(\displaystyle \left( {\frac{{\sqrt[3]{{1 + {\pi ^2}x}}}}{x}} \right)\left( {\frac{{{{(\sqrt[3]{{1 + {\pi ^2}x}})}^2} - (\sqrt[3]{{1 + {\pi ^2}x}}) + 1}}{{\sqrt[3]{{1 + {\pi ^2}x}}{)^2} - (\sqrt[3]{{1 + {\pi ^2}x}}) + 1}}} \right) = \frac{{1 + {\pi ^2}x - 1}}{{x\left( {\sqrt[3]{{1 + {\pi ^2}x}}{)^2} - (\sqrt[3]{{1 + {\pi ^2}x}}) + 1} \right)}}\)

how do i get rid of -1 in the numerator in the original formula?

 

[Deleted member 4993]

how do i get rid of -1 in the numerator in the original formula?

The solution provided by pka (#9) had a small typo. However,if you work with paper and pencil and follow the general guideline provided by him (and in earlier responses), you should be able to get there easily!!

 

[Cilva7]

The only method I can think of is L hoptals rule. All you do is take the derivative of the top and bottom and then try taking the limit.

 

[Steven G]

You do not understand the method above? Where did you get stuck?