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

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In summary, to simplify the expression cos^2x-1/x^2, you can use the trigonometric identity cos^2x = (1+cos2x)/2 and the common denominator method to combine fractions. You cannot cancel out the x^2 in the denominator, but you can use direct substitution or other methods to find the limit of the simplified expression. The domain of the expression is all real numbers except for x=0, and a calculator can be used to solve the limit question but may not always give the most accurate or simplified answer. It is recommended to use a combination of algebraic and calculator methods for limit questions.
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d.tran103
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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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  • #2
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'Hopitals rule. This limit will help you with the limit in the OP...
 

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

How do I simplify the expression cos^2x-1/x^2?

To simplify the given expression, you can use the trigonometric identity cos^2x = (1+cos2x)/2. This will result in the expression (1+cos2x)/2-1/x^2. Next, you can use the common denominator x^2 to combine the two fractions. This will give you the simplified expression (x^2+cos2x-2)/2x^2.

Can I cancel out the x^2 in the denominator?

No, you cannot cancel out the x^2 in the denominator as it is not a common factor in both terms. Instead, you can use the common denominator method as mentioned in the previous answer to simplify the expression.

How do I find the limit of the simplified expression?

To find the limit of the simplified expression, you can use the direct substitution method. This means substituting the given value of x into the expression and evaluating the result. If the expression is in an indeterminate form, you can use other methods such as L'Hopital's rule or trigonometric identities to simplify and find the limit.

What is the domain of this expression?

The domain of the given expression is all real numbers except for x=0, as the expression is undefined at this point due to division by zero.

Can I use a calculator to solve this limit question?

Yes, you can use a calculator to evaluate the limit of this expression. However, it is important to note the limitations of calculators, as they may not always give the most accurate or simplified answer. It is recommended to use a combination of algebraic and calculator methods to solve limit questions.

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