Solving Trigo Limit Type 0/0 using L'Hopital's Rule | Sin x - x / x - tan x

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Homework Help Overview

The problem involves finding the limit of the expression (sin x - x)/(x - tan x) as x approaches zero, which presents a 0/0 indeterminate form. The subject area is calculus, specifically focusing on limits and the application of L'Hopital's Rule.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the repeated application of L'Hopital's Rule, noting that it complicates the expression further. There are suggestions to simplify the expressions by rewriting secant and tangent in terms of sine and cosine. One participant questions how to determine when to stop differentiating and consider alternative methods.

Discussion Status

The discussion is ongoing, with participants sharing their attempts and questioning the effectiveness of their approaches. Some guidance has been offered regarding simplification strategies, but there is no explicit consensus on the best path forward.

Contextual Notes

Participants are working under the constraints of a typical homework assignment, which may limit the methods they can use or the depth of exploration allowed. The presence of the 0/0 form is a key aspect of the problem being discussed.

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Homework Statement


Find limit of (sin x - x)/(x - tan x) as x approaches zero



Homework Equations


Type 0/0 , use L'Hopital's rule, differentiate.



The Attempt at a Solution


Every time I apply the rule it gets more complicated

(cos x - 1)/(1 - sec^2 x)
(sin x)/(2 sec^2 x tan x)
(cos x)/(2 Sec^4 [x] - 4 Sec^2 [x] Tan^2 [x])

etc etc please help
 
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Deathfish said:

Homework Statement


Find limit of (sin x - x)/(x - tan x) as x approaches zero



Homework Equations


Type 0/0 , use L'Hopital's rule, differentiate.



The Attempt at a Solution


Every time I apply the rule it gets more complicated

(cos x - 1)/(1 - sec^2 x)
(sin x)/(2 sec^2 x tan x)

Stop here, write sec and tan in terms of sin and cos. Eliminate sin from numerator en denominator.
 
Ok usually how do you decide when to stop differentiation and use alternative method?
 
Deathfish said:
Ok usually how do you decide when to stop differentiation and use alternative method?

The idea is to simplify the formula enough after each differentiation. That way you can see that you need to stop differentiation. Here, you need to write everything in sin and cos. Then you'll see after the second differentiation that you can cancel thingies.
 

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