Question 1161387
{{{f(x)=(x^2-3x-4)/(x^2-16)}}}

{{{f(x)=((x + 1) (x - 4))/((x-4)(x+4))}}}

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 < -4}}}  or  {{{-4 < x < 4}}} or  {{{x > 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(x)=((x + 1) (x - 4))/((x-4)(x+4))}}}....simplify

{{{f(x)=(x + 1)/(x+4)}}}

find limit

{{{lim(x->4,(x + 1)/(x+4))=(4 + 1)/(4+4)=5/8}}}

a function IS continuous at point {{{x=4}}}


{{{ graph( 600, 600, -10, 10, -10, 10, (x^2-3x-4)/(x^2-16)) }}}