If f(x) is continuous everywhere and lim-->3 (f(x)+10) / (x-3) = 3

Question

If f(x) is everywhere continuous and it is given that

lim-->3 (f(x)+10) / (x-3) = 3

It follows that y = ________ is the equation of the tangent line to y=f(x) at the point (__,__)

Answer 1

Key calculus fact: the slope of the tangent line equals the derivative.

Now let’s look at the definition of the derivative:

f’(x) at x = a is: lim x—>a (f(x) – f(a)/(x – a)

Now notice the limit given in your problem is close to this formulation, with a being 3 and f(a) being -10. So the limit given tells us the slope of the tangent is 3 at x = 3 and f(3) = -10.

So the equation of the tangent line is y = 3x + b, where we just need b. Plug in x = 3, y = -10 to find b:

(-10) = 3(3) + b, so b = -19.

ANSWER: the tangent line has equation y = 3x – 19 at the point (3,-10).