How can I solve this limit algebraically without using the conjugate method?

Thread starter Schrodinger's Cat
Start date Aug 31, 2006
Tags

Limits

Click For Summary

SUMMARY

The forum discussion revolves around solving the limit as x approaches 1 for the expression (x^0.5 - x^2) / (1 - x^0.5). Participants explored various algebraic techniques, including long division and factoring, to simplify the expression without using the conjugate method. The final conclusion reached is that the limit evaluates to 3, achieved through careful manipulation of the numerator and denominator, particularly by factoring and canceling terms. Key techniques discussed include polynomial long division and rewriting expressions to facilitate cancellation.

PREREQUISITES

Understanding of limits in calculus
Familiarity with polynomial long division
Ability to factor algebraic expressions
Knowledge of manipulating fractional powers of x

NEXT STEPS

Study polynomial long division techniques in detail
Learn how to factor cubic expressions effectively
Explore advanced limit techniques, including L'Hôpital's Rule
Practice solving limits involving fractional powers and algebraic manipulation

USEFUL FOR

Students and educators in calculus, mathematicians focusing on algebraic limits, and anyone looking to enhance their skills in manipulating algebraic expressions for limit evaluation.

Sep 1, 2006

daveb

I can't read the jpeg for some reason, but I also got 3 by decomposing the numerator in the original equation (x^0.5-x^2) = x^0.5(1-(x^0.5)^3) which is a cubic and can be further factored as x^0.5(1-x^0.5)(1+x^0.5+x). The (1-x^0.5) cancel out leaving x^0.5)(1+x^0.5+x). Then take this limit as x goes to 1 gives 3. I imagine that's similar to what wurth did