find the value for which the function is continuous

• August 2nd 2010, 12:39 PM
dat1611
find the value for which the function is continuous
f(x)= ln x/(x-1), if 0<x not =1
c , if x=1
and what kind of discontinuity is present if c does not have this value?

can someone help me with this problem?
• August 2nd 2010, 04:40 PM
Prove It
This is nearly impossible to read...

Is it

$f(x) = \begin{cases}\frac{\ln{x}}{x - 1}\textrm{ if }x \in (0, 1)\cup (1, \infty)\\c\textrm{ if }x=1\end{cases}$

or

$f(x) = \begin{cases}\ln{\left(\frac{x}{x - 1}\right)}\textrm{ if }x \in (0, 1)\cup (1, \infty)\\c\textrm{ if }x=1\end{cases}$?
• August 3rd 2010, 12:44 PM
dat1611
$f(x) = \begin{cases}\frac{\ln{x}}{x - 1}\textrm{ if } 0

thats how it is i dont know how to do a not equal sign
• August 3rd 2010, 12:54 PM
skeeter
Quote:

Originally Posted by dat1611
$f(x) = \begin{cases}\frac{\ln{x}}{x - 1}\textrm{ if } x>0 \, ; \, x \ne 1 \\c\textrm{ if }x=1\end{cases}$

fify
• August 3rd 2010, 01:45 PM
Vlasev
Quote:

Originally Posted by dat1611
$f(x) = \begin{cases}\frac{\ln{x}}{x - 1}\textrm{ if } 0

thats how it is i dont know how to do a not equal sign

It is done by \neq

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For the question at hand, you need to determine a value c such that the function is continuous. One way could be to just plug in 1 in the above definition with the natural log, but you cannot do that since you'd get something of the form 0/0. Hence you need to find the limit of [LaTeX ERROR: Convert failed] as x approaches 1. Or more succinctly, find

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Once you find that, you can just set c equal to it. From the definitions of continuous functions, f(x) is continuous at point a if

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• August 3rd 2010, 01:46 PM
dat1611
ok can someone help with the actual problem
• August 3rd 2010, 01:56 PM
Vlasev
Do you know L'Hopital's rule? If you do, just apply to the numerator and denominator and it ' should be pretty easy.
• August 3rd 2010, 02:13 PM
Soroban
Hello, dat1611!

Quote:

$f(x)\;=\; \left\{ \begin{array}{cc}\dfrac{\ln x}{x-1}& x > 0,\;x \ne 1 \\ \\[-3mm]
c & x=1 \end{array} \right$

Find the value of $c$ for which the function is continuous.

To be continuous at $x = 1$, we want: . $\displaystyle{\lim_{x\to1}\frac{\ln x}{x-1} \;=\;c }$

The left side goes to $\frac{0}{0}$ so we can apply L'Hopital.

. $\displaystyle{\lim_{x\to1}\frac{\frac{1}{x}}{1} \;=\;c \quad\Rightarrow\quad \lim_{x\to1}\frac{1}{x} \:=\:c$

Therefore: . $c \:=\:1$

• August 3rd 2010, 02:21 PM
dat1611
so
$\displaystyle \lim\limit_{x \to 1}\frac{\ln{x}}{x - 1}
$
= 1