What is the value of c that makes f(x) continuous on the entire real line?

[hotrocks007]

Determine the value of c so that f(x) is continuous on the entire real line if

f(x)={ x^2, x<_3 (less than or equal to)
{ c/x, x>3

I've been trying by just guess and check, but I have absolutely NO idea!
Please help!

 

[lurflurf]

Determine the value of c so that f(x) is continuous on the entire real line if

f(x)={ x^2, x<_3 (less than or equal to)
{ c/x, x>3

I've been trying by just guess and check, but I have absolutely NO idea!
Please help!

Choose c so that
$$\lim_{x\rightarrow 3^-}f(x)=\lim_{x\rightarrow 3^+}f(x)=f(3)$$

 

[thepatientmental]

To determine the value of c, we need to make sure that the function is continuous at the point x=3. This means that the limit of the function as x approaches 3 from both the left and the right must be equal. In other words, the left and right limits must "meet" at x=3.

Taking the left limit, we have:

lim x→3- f(x) = lim x→3- x^2 = 3^2 = 9

Taking the right limit, we have:

lim x→3+ f(x) = lim x→3+ c/x = c/3

For the function to be continuous at x=3, these two limits must be equal. Therefore, we can set them equal to each other and solve for c:

9 = c/3

c = 27

So, the value of c that makes f(x) continuous on the entire real line is 27.