Question 1200312
<pre>

The question is badly worded, because unless a continuous function is a linear
function, there are infinitely many tangent lines, not just one tangent line as
the words "the tangent line" here seems to imply.

The question should be:

"From the given information, is there a point at which we can determine the
equation of the tangent line? If so what is that point and what is the equation
of the tangent line at that point."

{{{lim["x->4"]}}}{{{(f(x)+2)/(x-4)}}}{{{""=""}}}{{{11}}}

Since the denominator is 0 when x=4, the numerator must also = 0 when x=4.
Therefore,

f(4)+2 = 0
  f(4) = -2

Therefore, we know that (4,-2) is a point on f(x).

Since both the numerator and denominator of {{{(f(x)+2)/(x-4)}}} both
approach 0 as x approaches 4, we know that L'Hopital's rule holds.  Therefore
the quotient of the derivatives of the numerator and denominator has the same
limit 11 as x approaches 4.

{{{lim["x->4"]}}}{{{("f'(x)")/(1)}}}{{{""=""}}}{{{11}}}

Therefore, f'(4) = 11, the slope of a tangent line at (4,-2)

So we know that at the point (4,-2) the slope of the tangent line is 11.

Using the point-slope formula for the equation of a line:

{{{y-y[1]}}}{{{""=""}}}{{{m(x-x[1])}}}

{{{y-(-2)}}}{{{""=""}}}{{{11(x-4)}}}

{{{y+2}}}{{{""=""}}}{{{11x-44)}}}

{{{y}}}{{{""=""}}}{{{11x-46)}}} is the equation of the tangent line to

f(x) at the point (4,-2)

Edwin</pre>