Find this limit without L'Hopital's Rule

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SUMMARY

The limit lim (sqrt(x)/sqrt(sin(x))) as x approaches 0 from the positive side can be evaluated without L'Hopital's Rule by utilizing the power series expansion of sin(x). Specifically, the relevant limit is lim_{x -> 0} (sin(x)/x) = 1, which aids in simplifying the expression. The solution involves recognizing that as x approaches 0, sin(x) can be approximated by its Taylor series expansion, leading to a definitive evaluation of the limit.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with Taylor series expansions
  • Knowledge of the properties of sine function
  • Ability to manipulate algebraic expressions
NEXT STEPS
  • Study Taylor series expansion for sin(x) and its applications
  • Learn about alternative limit evaluation techniques beyond L'Hopital's Rule
  • Explore the concept of continuity in functions and its implications for limits
  • Review the properties of square roots in calculus
USEFUL FOR

Students of calculus, mathematics educators, and anyone looking to deepen their understanding of limit evaluation techniques without relying on L'Hopital's Rule.

htoor9
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Homework Statement



L'Hopital's rule does not help with this limit. Find it some other way. lim (squarert(x)/squarert(sinx)) as x -> 0+

Homework Equations



None?

The Attempt at a Solution



The only way I can think of solving this is by using L'Hopital's rule...but it obviously isn't working. How else can I find this limit?
 
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htoor9 said:

Homework Statement



L'Hopital's rule does not help with this limit. Find it some other way. lim (squarert(x)/squarert(sinx)) as x -> 0+

Homework Equations



None?
There are at least a couple of limits that are relevant.
[tex]\lim_{x \to 0} \frac{sin(x)}{x} = 1[/tex]
[tex]\lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x)), \text{provided that f is continuous at g(a)}[/tex]
htoor9 said:

The Attempt at a Solution



The only way I can think of solving this is by using L'Hopital's rule...but it obviously isn't working. How else can I find this limit?
 
Expand sin x as a power series...
 

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