It should be -7x² instead of -7x. And the right hand side expression cannot be divided by x³ throughout as there's no denominator unlike x²/(x - 8). You would be reducing the limit this way.

How you showing me the correct working for analysis??

From your 3rd last step :

(1 + lim ( (x²/x²)/(2/x - 8/x²) )^⅓ ) ( lim(x³ - 7x² - 4) )

= (1 + (1/(2/8 - 8/8²))^⅓ ) (8³ - 7(8)² - 4)

= (1 + (1/⅛)^⅓ ) (512 - 448 - 4)

= (1 + 8^⅓)(60)

= (1 + 2)(60)
= 3 x 60
= 180

J : Do you have alternative solution to this question?

I got 180 as well, can you take a look at my ans and see if I presented correctly?

Congrat Noel! Most important is that your underlying principle must be correct.

Your working seems fine to me! But maybe it can be clearer by adding an additional step to show the evaluation of the limit for the cube root, since you've shown it for the right hand side term as well.

eg. The left hand side

(1 + ³√(lim x² / lim(2x - 8) ) )

= (1 + ³√ (8²/ (2(8) - 8) )

= (1 + ³√8)

= (1 + 2)

ok thanks! :) I have another limit proving question, but idk how to present the ans using the squeeze theorem or precise definition of a limit, is it ok if you help me take a look?

I can only try! Kind of rusty at all these tbh