Making a Function Continuous

Question

Find the values of a and b that make the function f(x) continuous if:

 

f(x)=sinx/x for x less than zero

ax+b for x greater than or equal to zero and less than or equal to two

x^2+3 for x greater than two

 

I tried finding the limit as x approaches zero from the left, and got one. The limit as x approaches two from the right is seven. How do I plug these numbers into ax+b to find the equation for this line that would make it continuous?

Thank you!

 

Answer 1

Hi Noelle,

 

OK to solve this problem, we need to do two things:

 

First, we need to know what the limit as x approaches 0 of sin(x)/x is, and second what the limit as x approaches 2 of x²+3 is.

 

The limit as x approaches 0 of sin(x)/x is of indeterminate form, so we can use l'Hoptial's rule. The rule tells us that the limit as x approaches 0 of sin(x)/x is the same as the limit of the derivatives, i.e. the limit as x approaches 0 of cos(x)/1, which is just cos(0)/1=1. So, the limit as x approaches the 0 of sin(x)/x is 1.

 

The limit as x approaches 2 of x²+3 is just 2²+3=7.

 

Now, we know the endpoints, so we can go ahead and solve for a and b.

 

We know that the left endpoint for when x=0 should give us a 1, so a*0+b=1, i.e. b=1.

 

We also know that the right endpoint for when x=2 is 7, so a(2)+1-7, or a=3.

 

Hence, the values a=3 and b=1 will make the function continuous.