SOLUTION: Consider: f(x)=(x^2-3x-4)/(x^2-16) Give the domain of f(x). Test all the conditions of continuity to determine whether the given function is continuous of discontinuous at p

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Question 1161387: Consider:
f(x)=(x^2-3x-4)/(x^2-16)
Give the domain of f(x).
Test all the conditions of continuity to determine whether the given function is continuous of discontinuous at point x=4.
Found 3 solutions by MathLover1, solver91311, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=%28x%5E2-3x-4%29%2F%28x%5E2-16%29
f%28x%29=%28%28x+%2B+1%29+%28x+-+4%29%29%2F%28%28x-4%29%28x%2B4%29%29
The function domain:
all values of x except those which make denominator equal to zero and these are:
x=4 and x=-4
so,
x+%3C+-4 or -4+%3C+x+%3C+4 or x+%3E+4
interval notation:
(-infinity+,-4) U (-4,4) U (4,infinity+)
to determine whether a function is continuous at point x=-4:
since x=-4 is excluded from domain because it makes 0 in the denominator, a function is NOT continuous at point x=-4

to determine whether a function is continuous at point x=4 check the limit:
the limit of the function as x approaches the value 4 must exist
f%28x%29=%28%28x+%2B+1%29+%28x+-+4%29%29%2F%28%28x-4%29%28x%2B4%29%29....simplify
f%28x%29=%28x+%2B+1%29%2F%28x%2B4%29
find limit
lim%28x-%3E4%2C%28x+%2B+1%29%2F%28x%2B4%29%29=%284+%2B+1%29%2F%284%2B4%29=5%2F8
a function IS continuous at point x=4

+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%28x%5E2-3x-4%29%2F%28x%5E2-16%29%29+


Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!




A function is continuous at a point if and only if .

So evaluate

and evaluate .

If they are the same, the function is continuous.


John

My calculator said it, I believe it, that settles it

Answer by ikleyn(52797) About Me  (Show Source):
You can put this solution on YOUR website!
.

Considering responses by tutor  @solver91311 and tutor  @MathLover1,  NO  ONE  of them is correct.

The given function IS NOT defined at  x= 4.

The number  "4"  is not a part of the domain to this function.

Therefore,  the function  f(x)  IS  NOT  CONTINUOUS  at this point.


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It is a standard exam question to check if the student knows the definition of the continuity of the function at the given point.

So, be aware !