Please provide the reason of your confusion ...KevinFan said:For what value of the constant c is the following function continuous at x = −3?
f(x)=(1/x+1/3)/(x+3) if x≠ -3
f(x)=c if x=-3please provide proof... I am so confused:(
I am confused because the second part of the function is just "c" which is a constant does not involve x.fresh_42 said:Please provide the reason of your confusion ...
I have tried to simplfy the function, it is f(x)=1/(3x) and I noticed for the simplfied function, x can be -3fresh_42 said:What happens with ##f(x)## for ##x \neq -3## if you simplify the fractions? Have you tried to draw a graph around ##x = -3##?
oh, is C=1/(3(-3))=-1/9 ??fresh_42 said:What happens with ##f(x)## for ##x \neq -3## if you simplify the fractions? Have you tried to draw a graph around ##x = -3##?
Yes. So now you have to find an argument, why ##f(x)## becomes continuous if we set ##f(-3)= -\frac{1}{9}##.KevinFan said:oh, is C=1/(3(-3))=-1/9 ??
Many thanks for your help !fresh_42 said:Yes. So now you have to find an argument, why ##f(x)## becomes continuous if we set ##f(-3)= -\frac{1}{9}##.
On the oringinal function when x= -3, the function is undefined. However, if we set f(-3)=-1/9 then the oringinal function will become continuous on x=-3.fresh_42 said:Yes. So now you have to find an argument, why ##f(x)## becomes continuous if we set ##f(-3)= -\frac{1}{9}##.
A function is continuous at x=-3 if the limit of the function as x approaches -3 exists and is equal to the value of the function at x=-3. This means that there are no sudden jumps or breaks in the graph of the function at x=-3.
To prove that a function is continuous at x=-3, you must show that the limit of the function as x approaches -3 exists and is equal to the value of the function at x=-3. This can be done using the definition of continuity and various limit theorems.
The value of x=-3 is important because it is the point at which we are evaluating the continuity of the function. In order for a function to be continuous at x=-3, the limit of the function as x approaches -3 must exist and be equal to the value of the function at x=-3.
Yes, a function can be continuous at x=-3 but not at other points. This means that the function may have sudden jumps or breaks at other points, but at x=-3, the limit of the function exists and is equal to the value of the function at x=-3.
Some common techniques for proving continuity at x=-3 include using the definition of continuity, using the limit laws and properties, and using the intermediate value theorem. It is important to carefully analyze the function and its behavior at x=-3 in order to determine the best approach for proving continuity.