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limit and graph

aruwin

Member
Jul 4, 2012
121
I have to find the limit of a function and draw a graph of it. First, can you check my work and then tell me how to draw the graph?


When 2 variables x,y are related by y=1+xlogy, find lim(x->1) dy/dx.

So,my first step is differentiating both sides:
dy/dx = logy + x*(1/y)*(dy/dx)
dy/dx = (y*logy)/(y - x)

So, lim(x->1) dy/dx = lim(x->1) (y*logy)/(y - x) = y*logy/(y-1)

Did I get the limit right? So how to draw the graph?
 

chisigma

Well-known member
Feb 13, 2012
1,704
I have to find the limit of a function and draw a graph of it. First, can you check my work and then tell me how to draw the graph?


When 2 variables x,y are related by y=1+xlogy, find lim(x->1) dy/dx.

So,my first step is differentiating both sides:
dy/dx = logy + x*(1/y)*(dy/dx)
dy/dx = (y*logy)/(y - x)

So, lim(x->1) dy/dx = lim(x->1) (y*logy)/(y - x) = y*logy/(y-1)

Did I get the limit right? So how to draw the graph?
We have a function y=f(x) implicity defined as...

$\displaystyle g(x,y)= y -x\ \ln y -1 =0$ (1)

... and, according to the Dini's Theorem, its derivative is ...

$\displaystyle f^{\ '} (x)= - \frac{g^{\ '}_{x}(x.y)}{g^{\ '}_{y}(x,y)}= \frac{\ln y}{1-\frac{x}{y}}$ (2)

For x=1 the (1) has the solution y=1 so that [applying l'Hopital's rule...] is...

$\displaystyle \lim_{x \rightarrow 1} f^{\ '}(x) = \lim_{y \rightarrow 1} \frac{y\ \ln y}{y-1} = \lim_{y \rightarrow 1} (1+\ln y)=1$ (3)

Kind regards

$\chi$ $\sigma$
 

aruwin

Member
Jul 4, 2012
121
We have a function y=f(x) implicity defined as...

$\displaystyle g(x,y)= y -x\ \ln y -1 =0$ (1)

... and, according to the Dini's Theorem, its derivative is ...

$\displaystyle f^{\ '} (x)= - \frac{g^{\ '}_{x}(x.y)}{g^{\ '}_{y}(x,y)}= \frac{\ln y}{1-\frac{x}{y}}$ (2)

For x=1 the (1) has the solution y=1 so that [applying l'Hopital's rule...] is...

$\displaystyle \lim_{x \rightarrow 1} f^{\ '}(x) = \lim_{y \rightarrow 1} \frac{y\ \ln y}{y-1} = \lim_{y \rightarrow 1} (1+\ln y)=1$ (3)

Kind regards

$\chi$ $\sigma$
Got it :) Now how to plot the graph? What sould I do first?
 

chisigma

Well-known member
Feb 13, 2012
1,704

aruwin

Member
Jul 4, 2012
121

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
I have to find the limit of a function and draw a graph of it. First, can you check my work and then tell me how to draw the graph?


When 2 variables x,y are related by y=1+xlogy, find lim(x->1) dy/dx.

So,my first step is differentiating both sides:
dy/dx = logy + x*(1/y)*(dy/dx)
dy/dx = (y*logy)/(y - x)

So, lim(x->1) dy/dx = lim(x->1) (y*logy)/(y - x) = y*logy/(y-1)

Did I get the limit right? So how to draw the graph?
Well, the obviousd thing to do is to write the equation as [tex]x= \frac{y- 1}{log y}[/tex]. Choose a number of different values of y, calculate the corresponding value of x, and plot (x, y). Once you get sufficient points, draw a smooth curve through them.
(You say to begin with that you "have to find the limit of a function" but then say "find lim(x->1) dy/dx". Are you to find the limit of the given function or its derivative?)
 

aruwin

Member
Jul 4, 2012
121
Well, the obviousd thing to do is to write the equation as [tex]x= \frac{y- 1}{log y}[/tex]. Choose a number of different values of y, calculate the corresponding value of x, and plot (x, y). Once you get sufficient points, draw a smooth curve through them.
(You say to begin with that you "have to find the limit of a function" but then say "find lim(x->1) dy/dx". Are you to find the limit of the given function or its derivative?)
Its derivative.