Solving Basic Limit Question: Cos^2x-1/x^2

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SUMMARY

The limit problem presented is lim (x→0) (cos²x - 1)/x², which evaluates to -1. The transformation of cos²x - 1 into -sin²x is crucial for simplifying the expression. Utilizing the limit lim (x→0) (sin(x)/x) is essential for solving this problem. Additionally, applying L'Hôpital's rule can effectively resolve the limit when direct substitution leads to an indeterminate form.

PREREQUISITES
  • Understanding of trigonometric limits, specifically lim (x→0) (sin(x)/x)
  • Familiarity with L'Hôpital's rule for evaluating indeterminate forms
  • Knowledge of trigonometric identities, particularly cos²x - 1 = -sin²x
  • Basic calculus concepts, including limits and continuity
NEXT STEPS
  • Study the application of L'Hôpital's rule in various limit problems
  • Explore trigonometric limits and their derivations
  • Practice simplifying expressions using trigonometric identities
  • Investigate advanced limit techniques, including Taylor series expansions
USEFUL FOR

Students studying calculus, mathematics educators, and anyone looking to deepen their understanding of limits and trigonometric functions.

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I'm stuck on this one problem:
lim (x=>0) cos^2x-1/x^2

I know the answer is -1, however I'm having trouble getting there.

Can anyone help me with this? I know cos^2x-1 is also -sin^2x and I'm having trouble with getting rid of the denominator. Thanks!
 
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So you know the following limit??

[tex]\lim_{x\rightarrow 0}{\frac{\sin(x)}{x}}[/tex]

??

If you don't know it, then you can calculate it with l'hospital's rule. This limit will help you with the limit in the OP...
 

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