Calculus: Finding the equation of the tangent line

Question

Assume that f(x) is everywhere continuous and it is given you to that lim x→5 (f(x)+4)/(x−5) = −8

it follows that y = _____ is the equation of the tangent line to y = f(x) at the point (_____, _____).

Answer 1

The only way the limit has a value when the denominator is approaching zero

is for the numerator to approach zero so the limit is in indeterminate 0/0 form

which can be resolved.

That is, f(x)=-4 when x=5

so point of tangency is (5,-4)

-8=(y+4)/(x-5)

y= -8x+36

Answer 2

f(x) + 4 = -8(x-5)

f(x) = -8x +40 -4

f(x) = -8x +36

f'(x) = -8 = the slope of the tangent line at any point on the line, such as (0, 36) the y intercept

y = -8x + 36 is the "tangent" line with slope = -8, through any point on the given line.

but that's strange terminology, as the "tangent" line is the same line as the given line

"tangent line" is a phrase referring to a line that just touches a curve, not a line that is the very same line. the "tangent line" is a straight line. the "curve" it just touches is not a straight line.

so, just make f(x)= some unknown non-linear function

then y=-8x+36 is the tangent line to f(x) at a point (x, f(x)), but we don't know what f(x) is, snut use (0, 36) as the y intercept of f(x) and of the tangent line.

BUT one point where there is no tangent line is (5,-4) which is not even on the line. It's a "hole" in the line, with no tangent line "touching" that point, as that point is not on the line or curve f(x)