Calculus Twice Differentiable Function Tangent Line Equation Problem Help

Question

Question: Suppose that $f(x)$ is a twice-differentiable function such that $\lim_{x\to 3}\dfrac{12x-36}{3f(x)-6}=-1$. Find the equation of the tangent line to $f(x)$ at $x=3$.

So I know a couple things: because f(x) has to be twice-differentiable, it must be to the ^2. Another thing I know is that since the derivative of th top is 12, the denominator has to be negative (or at least, 6*(something) has to be negative -12). Is the answer really as simple as -2, so f(x) is -2x^2 ? It can't be that simple because you also have to incorporate the product rule right? May I please have some help? I'm stuck. Thanks!

Answer 1

$f(x)$ doesn't necessarily have to be any polynomial or any specific kind of function.

Since the numerator equals $0$, the limit equal $-1$ then the denominator must equal $0$

$f(3) = 2$

And since both numerator and denominator equal 0 we can apply L'Hopital's rule.

$\lim_\limits{x\to 3} \frac {12x - 36}{3f(x) - 6} = \lim_\limits{x\to 3} \frac {\frac{d}{dx}(12x - 36)}{\frac {d}{dx} 3f(x) - 6} = \frac {12}{3f'(3)} = -1$

$f'(3) = -4$

And now since you know $f(3)$ and $f'(3)$ you can find the equation of the line tangent to $f(x)$ at $x = 3$

$(y-2) = -4(x-3)$

Answer 2

We are given $\lim_{x\to3} \frac{12x-36}{3f(x)-6} = -1$. Simplify by dividing out the common factor, 3.

$$\lim_{x\to3} \frac{4x-12}{f(x)-2} = -1$$ You can pull a common factor out of a multiple: lim C*L = C*lim L, C is a constant, L is a function. $$4\cdot\lim_{x\to3} \frac{x-3}{f(x)-2} = -1$$ $$\lim_{x\to3} \frac{x-3}{f(x)-2} = -\frac14$$ The limit of a fraction is equal to its reciprocal: if lim A/B = C, then lim B/A = 1/C. $$\lim_{x\to3} \frac{f(x)-2}{x-3} = -4$$

The not-so-common definition of a tangent slope is $m = \lim_{x\to a} \frac{f(x)-f(a)}{x-a}$.

If we just plug the limit equation in to the slope formula identify variables and etc., we get that $a=3, f(a) = 2,$ and $f'(a) = -4$. Use point-slope formula now: $y-y_0 = m(x-x_0)$ --> $$y-2 = -4(x-3)$$ $$y=-4x+14$$