SOLUTION: For what value(s) of a and b is the function f(x) given below continuous at 0? {{{matrix(1,3, "f(x)",""="", system( matrix(3,3,sin(3x)/x, if, x<0, a, if, x=0, (5-sqrt(25-x

Algebra ->  Equations -> SOLUTION: For what value(s) of a and b is the function f(x) given below continuous at 0? {{{matrix(1,3, "f(x)",""="", system( matrix(3,3,sin(3x)/x, if, x<0, a, if, x=0, (5-sqrt(25-x      Log On


   

Question 1200105: For what value(s) of a and b is the function f(x) given below continuous at 0?

Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!


The 3 parts of the graphs are 

y=sin%283x%29%2Fx, y=a, y=%285-sqrt%2825-x%29%29%2F%28bx%29

The first function must approach the same value as the second function as x approaches 0 from the left. So

lim%5B%22x-%3E0-%22%5D%28sin%283x%29%2Fx%29%22%22=%22%22lim%5B%22x-%3E0%22%5D3%2A%28sin%283x%29%2F%283x%29%29%22%22=%22%223%2Alim%5B%22x-%3E0%22%5D%28sin%283x%29%2F%283x%29%29%22%22=%22%223%2A%281%29%22%22=%22%223

Therefore a = 3, so the second function is y=3

Also the third function must approach the same value as the second function as
x approaches 0 from the right.

lim%5B%22x-%3E0%2B%22%5D%28%285-sqrt%2825-x%29%29%2F%28bx%29%29%22%22=%22%22lim%5B%22x-%3E0%2B%22%5D%22%22=%22%22
lim%5B%22x-%3E0%2B%22%5D%28%2825-%2825-x%29%29%2F%28bx%285%2Bsqrt%2825-x%29%29%29%29%29%22%22=%22%22lim%5B%22x-%3E0%2B%22%5D%2825-25%2Bx%29%2F%28bx%285%2Bsqrt%2825-x%29%29%29%29%29%22%22=%22%22lim%5B%22x-%3E0%2B%22%5D%28x%29%2F%28bx%285%2Bsqrt%2825-x%29%29%29%29%29%22%22=%22%22
lim%5B%22x-%3E0%2B%22%5Dcross%28x%29%2F%28b%2Across%28x%29%285%2Bsqrt%2825-x%29%29%29%29%29%22%22=%22%22lim%5B%22x-%3E0%2B%22%5D1%2F%28b%2A%285%2Bsqrt%2825-x%29%29%29%29%29%22%22=%22%221%2F%28b%2A%285%2Bsqrt%2825-0%29%29%29%29%29%22%22=%22%221%2F%28b%2A%285%2Bsqrt%2825%29%29%29%29%29%22%22=%22%221%2F%28b%2A%285%2B5%29%29%29%29%22%22=%22%221%2F%2810b%29

So 1%2F%2810b%29%22%22=%22%223

1%22%22=%22%2230b

1%2F30%22%22=%22%22b

Here's the graph of the continuous function:



Edwin

Answer by ikleyn(52848) About Me  (Show Source):
You can put this solution on YOUR website!
.

Look at this web-site

https://medium.com/@nadzeya/the-wonderful-limits-in-maths-part-1-94c6b41a80e8


(free of charge), and find there the proof of one of the wonderful limits


    sin%28x%29%2Fx  ---> 1  as  x ---> 0.


It is one of wonderful limits from Calculus that any Calculus student must know.