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evaluate the limit.$$\lim _{x \rightarrow 0}\left(\cot x-\frac{1}{x}\right)$$

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Calculus 1 / AB

Chapter 4

APPLICATIONS OF THE DERIVATIVE

Section 5

L'Hopital's Rule

Derivatives

Differentiation

Applications of the Derivative

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University of Nottingham

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Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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For this question, we're going to evaluate the limit and what we can do is use hopital to combine this into x, co, tangent, x, minus 1 over x, and if we take our limit expression 0 we're going to get infinity over infinity, which is indeterminate. We can use a petal, so the derivative of f, which is going to be numerator g being a denominator, is going to be x, negative, x, co, secant squared x, plus co tangent x, while the derivative of g is just going to be 1, so we can Rewrite our limit as a limit of x, approaching 0 of co, tangent x minus x. Now what we can do. We can also rewrite this with sin denominator. So we'll do co. Tangent of x, minus x, cosecant square of x can be rewritten as co sine x over sine x minus x over sine squared x. So if we make this sine squared x, we multiply sine here. We write this as sine x, cosine x, minus x, over sine squared x. Now, taking the limit of this, we still can't do it we're going to get easier in the denominator, we'll differentiate again so in the bottom, we're going to get the derivative of sine squared x is just going to be 2 sine x, cosine x, derivative of sine X, cosine x, we can actually do a trick with trig and place 1 half sine of 2 x. Take that thrives going to give us a cosine of 2 x derivative of x is 1 to minus 1 point. So the limit of this will still give us a determinate form so 1 more thing we to do. We can make this sine of 2 x and differentiate again, so derivative cosine 2 x is just going to be negative. 2 sine of 2 x derivative of sine of 2 x is going to give us 2 cosine 2 plugging in our x approaching 0. Now we're going to see successfully we're going to get 0 numerator but 1 in the denominator. So our limit evaluates out to 0.

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