1. **Problem Statement:** We want to understand the concept of continuity and find the slope of the tangent line to a function at a point, using an intuitive example. 2. **Continuity Definition:** A function $f(x)$ is continuous at a point $x=a$ if three conditions hold: - $f(a)$ is defined. - The limit $\lim_{x \to a} f(x)$ exists. - The limit equals the function value: $\lim_{x \to a} f(x) = f(a)$. 3. **Tangent Slope Definition:** The slope of the tangent line to $f(x)$ at $x=a$ is the derivative $f'(a)$, defined as: $$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$ This limit represents the instantaneous rate of change of $f$ at $a$. 4. **Example Function:** Consider $f(x) = x^2$. 5. **Check Continuity at $x=2$:** - $f(2) = 2^2 = 4$. - $\lim_{x \to 2} x^2 = 4$ since $x^2$ is continuous everywhere. - Since the limit equals the function value, $f$ is continuous at $x=2$. 6. **Find Tangent Slope at $x=2$:** Calculate the derivative using the limit definition: $$f'(2) = \lim_{h \to 0} \frac{(2+h)^2 - 2^2}{h} = \lim_{h \to 0} \frac{4 + 4h + h^2 - 4}{h} = \lim_{h \to 0} \frac{4h + h^2}{h}$$ Simplify the fraction: $$= \lim_{h \to 0} (4 + h) = 4$$ 7. **Interpretation:** The function $f(x) = x^2$ is smooth and continuous at $x=2$, and the slope of the tangent line there is 4. This means the instantaneous rate of change of $f$ at $x=2$ is 4. **Final answer:** - $f$ is continuous at $x=2$. - The slope of the tangent line at $x=2$ is $4$.