GATE CSE
First time here? Checkout the FAQ!
x
+1 vote
132 views

What should be the value of a,b and c such that the function defined below is continuous at x=0 ?

                   
                            $f\left ( x \right )=\begin{Bmatrix} \left ( 1+ax \right )^{\frac{1}{x}} & x<0 & \\ b & x=0& \\ \frac{(x+c)^{\frac{1}{3}}-1}{x}                     &x>0 & \end{Bmatrix}$

asked in Calculus by Veteran (35.4k points)   | 132 views
a = b = infinity, if c = any finite value other than 1??

If c = 1, then , b = 1/3 , a = ln (1/3).

??
Yes right.Provide the solution by considering second case.

See.. Since it is continuous at x = 0, so f(0-) = f(0+) = f(0). ------- (X)

Here f(0-) = Lim(x->0) { (1+ax)1/x } = ea .

        f(0)  = b

and  f(0+) =Lim ( x->0)  {(x + c)1/3 - 1}/ (x)

Now if c = 1 then,

         f(0+) =Lim ( x->0) {(x + 1)1/3 - 1} /(x)  => 0/0

  using L hospital rule, we can solve above limit = 1/3 .

Hence from eqn (X) , ea = b = 1/3.

  a = ln (1/3)

  b = 1/3

  c = 1.

Is it okay now  ?

yes.

1 Answer

0 votes
$a= $any value

$b=1$

$c$=$x^{3}+2x+3x^{2}+1$

Then x will be continuous at x=1

But continuous in x=0 not possible here
answered by Veteran (49.8k points)  
No .Actually we have given that  the above function is continuous at x=0 so  find the value of a,b,c.
Top Users Jan 2017
  1. Debashish Deka

    8978 Points

  2. sudsho

    5326 Points

  3. Habibkhan

    4798 Points

  4. Bikram

    4676 Points

  5. Vijay Thakur

    4496 Points

  6. saurabh rai

    4222 Points

  7. Arjun

    4216 Points

  8. Sushant Gokhale

    3834 Points

  9. santhoshdevulapally

    3808 Points

  10. Kapil

    3736 Points

Monthly Topper: Rs. 500 gift card

19,229 questions
24,123 answers
53,256 comments
20,323 users