Student Question

a) Find dy/dx , x^y=y^x

b) Use L'Hospitals's Rule to find the limit

lim x--> 1 (1-x)/(sinpix- lnx)

 

 

 

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Notice that the function from denominator is `sin pi*x - ln x`  and the function from numerator is `1 - x,`  hence, if you substitute 1 for x to numerator and denominator yields:

`lim_(x-gt1) (1-x)/(sin pi*x - ln x) = (1-1)/(sin pi - ln 1) = 0/(0-0) = 0/0`

You should use the l'Hospital's theorem, hence you need to differentiate the numerator and denominator with respect to x, as independent functions, such that:

`lim_(x-gt1) ((1-x)')/((sin pi*x - ln x)') = lim_(x-gt1) (-1)/(pi*cos pi*x - 1/x)`

You need to substitute 1 for x such that:

`lim_(x-gt1) (-1)/(pi*cos pi*x - 1/x) = (-1)/(pi*cos pi - 1)`

`lim_(x-gt1) (-1)/(pi*cos pi*x - 1/x) = (-1)/(-pi-1)`

`lim_(x-gt1) (-1)/(pi*cos pi*x - 1/x) = 1/(pi+1) `

Hence, evaluating the limit of the function `(1-x)/(sin pi*x - ln x)`  yields `lim_(x-gt1) (1-x)/(sin pi*x - ln x) = 1/(pi+1).`

Approved by eNotes Editorial
An illustration of the letter 'A' in a speech bubbles

a)

`x^y = y^x`

using implicit differentiation,

`yx^(y-1) = xy^((x-1))*(dy)/(dx)`

`(dy)/(dx) = (yx^(y-1))/(xy^(x-1))`

`(dy)/(dx) = (x^(y-2))/(y^(x-2))`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Approved by eNotes Editorial