1. The problem is to understand and use the formula for the slope of the tangent line to a curve at a point, given by the limit: $$m = \lim_{x \to x_0} \frac{y - y_0}{x - x_0}$$ 2. This formula represents the derivative of the function at the point $x_0$, where $(x_0, y_0)$ is a point on the curve. 3. The slope $m$ is the limit of the average rate of change of $y$ with respect to $x$ as $x$ approaches $x_0$. 4. To use this formula, you substitute the function $y = f(x)$ and the point $(x_0, y_0)$, then simplify the difference quotient and take the limit as $x$ approaches $x_0$. 5. For example, if $y = f(x)$, then: $$m = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}$$ 6. This is the definition of the derivative $f'(x_0)$. 7. The key rule is to simplify the numerator to cancel the denominator factor $(x - x_0)$ before taking the limit. 8. If the limit exists and is finite, it gives the slope of the tangent line at $x_0$. This formula is fundamental in calculus for finding instantaneous rates of change.