Find a & b for Removable Continuity f(x)

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In summary, the conversation discusses finding the values of a and b in order for the function f(x) to be continuous for all values of x. The method of evaluating limits is mentioned, along with the definition of continuity and how it applies to the given function. The importance of finding the limits at specific values of x is also emphasized. The conversation concludes with the realization that a and b cannot be determined from the given information alone.
  • #1
jwxie
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This is from my old exam

f(x) = for x <1 (x-1)^2
for 1 <= x <= 4 ax+b find a and b so that fx is continuous for all x
for x <4 sqrt (2x+1)

so i guess i start evaluating some limit.

since the ax+b is define everywhere b/w x = 1 and x = 4, i guess i would use the third function f(x) = sqrt(2x+1) for limit goes to 4

f(x) = limit (x goes to 4) sqrt(s2+1) = 3
then i think i would let a = 3 ?
3x+b = ? what is y then?
 
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  • #2
Where might this function fail to be continuous?
 
  • #3
x = 4 and x = 1
 
  • #4
Yes, so what does it mean for this function to be continuous at x = 1 and x = 4?
 
  • #5
oh so the ax +b = y
since a = 3, for x = 4, y use the third equation which gives 9
3x+b = 9
am i correct
 
  • #6
Why does a = 3? The definition of continuity is: f is continuous at c if [tex]\lim_{ x \to c } f(x) = f(c)[/tex].
 
  • #7
for x <4 sqrt (2x+1)

Shouldn't it be :

for x > 4 : sqrt(2x+1)
 
  • #8
Yes, as jeques says, you must mean that [itex]f(x)= \sqrt{2x+1}[/itex] for x> 4.

What is the limit of f as x goes to 1 from below?
What is the limit of f as x goes to 1 from above?

For f to be continuous there, those two limits must be the same.

What is the limit of f as x goes to 4 from below?
What is the limit of f as x goes to 4 from above?

For f to be continuous there, those two limits must be the same.

You cannot determine a or b from either of those alone. Each gives an equation in both a and b.
 

Related to Find a & b for Removable Continuity f(x)

What is a removable discontinuity in a function?

A removable discontinuity, also known as a point discontinuity, is a type of discontinuity that occurs when there is a point in the domain of a function where the limit exists but the value of the function does not. This means that the function has a hole or gap at that point.

Why is it important to find a and b for removable continuity?

Finding the values of a and b is important for determining the type of discontinuity at a given point in a function. It also helps in identifying the behavior of the function near that point and understanding the overall continuity of the function.

How do you find a and b for removable continuity?

To find the values of a and b, you will need to use the limit definition of a derivative. This involves taking the limit of the function as it approaches the point of discontinuity and equating it to a and b. You can also use graphical methods or algebraic techniques to find these values.

Can a removable discontinuity be fixed?

Yes, a removable discontinuity can be fixed by redefining the function at that point. This can be done by filling in the hole or gap in the graph with the correct value. Once this is done, the function will become continuous at that point.

What are some real-life applications of finding a and b for removable continuity?

Finding the values of a and b for removable continuity is useful in various fields such as physics, engineering, and economics. It can help in analyzing the behavior of systems and predicting their future behavior. It is also used in optimization problems to find the maximum or minimum values of a function.

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