# Limit of x*cot(x)-1/x^2 as x->0

• tsuwal
In summary, the limit of x*cot(x)-1/x^2 as x approaches 0 is an indeterminacy that can be solved using l'Hopital's rule or by rearranging the expression and using the power series expansions of tan(x), sin(x), and cos(x). This can lead to a simpler solution for the limit.
tsuwal

## Homework Statement

limit x*cot(x)-1/x^2
x->0

lim x*cot(x)=1
x->0

## The Attempt at a Solution

it's an indetermination but L'hopital rule doesn't help very much..

tsuwal said:

## Homework Statement

limit x*cot(x)-1/x^2
x->0

lim x*cot(x)=1
x->0

## The Attempt at a Solution

it's an indetermination but L'hopital rule doesn't help very much..

Sometimes l'Hopital works better if you try rearranging things. Substitute cot(x)=cos(x)/sin(x). Multiply numerator and denominator by sin(x) and try again.

tsuwal said:
limit x*cot(x)-1/x^2
x->0
I assume you mean limit (x*cot(x)-1)/x2 as x->0.
The indeterminacy simply means you've not gone far enough in the expansion of the functions as power series. What expansions do you know for tan x, or failing that, for sin x and cos x?

thanks Dick, i wish i could see the those simplifications right away!

## 1. What is the limit of x*cot(x)-1/x^2 as x approaches 0?

The limit of x*cot(x)-1/x^2 as x approaches 0 is undefined. This can be seen by plugging in 0 for x, which results in an indeterminate form of 0/0. Therefore, other methods such as L'Hopital's rule or graphing may be used to evaluate the limit.

## 2. How is the limit of x*cot(x)-1/x^2 as x approaches 0 related to trigonometric functions?

The limit involves the cotangent function, which is the reciprocal of the tangent function. As x approaches 0, the value of cot(x) becomes very large, resulting in an undefined limit. This is due to the behavior of tangent approaching infinity as it approaches 0.

## 3. Can the limit of x*cot(x)-1/x^2 as x approaches 0 be evaluated using a calculator?

No, a calculator cannot evaluate this limit as it involves an indeterminate form. A calculator can only give an approximation of the limit by plugging in values close to 0, but this does not provide an exact answer.

## 4. What is the significance of the limit of x*cot(x)-1/x^2 as x approaches 0 in mathematics?

This limit is important in calculus as it demonstrates the concept of an indeterminate form and the need for other methods to evaluate such limits. It also highlights the behavior of trigonometric functions as they approach certain values.

## 5. How can the limit of x*cot(x)-1/x^2 as x approaches 0 be used in real-life applications?

This limit may be used in engineering or physics to model the behavior of certain systems or phenomena. For example, it could represent the rate of change of a quantity over time as it approaches a specific value. It could also be used in financial calculations to analyze the behavior of investments or interest rates.

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