Questions: Use the definition of continuity and the properties of limits to show that the function is continuous at the given number a. f(x) = 3x^4 - 5x + ∛(x^2+4), a=2

Solution Steps

To show that a function is continuous at a given point $a$, we need to verify that the limit of the function as $x$ approaches $a$ is equal to the function's value at $a$. This involves checking that the limit from the left and the right both equal $f(a)$.

1. Calculate $f(a)$.
2. Compute the limit of $f(x)$ as $x$ approaches $a$.
3. Verify that the limit equals $f(a)$.

To find the value of the function at $a = 2$, we compute: $$f(2) = 3(2)^4 - 5(2) + \sqrt[3]{(2)^2 + 4}$$ Calculating this gives: $$f(2) = 3(16) - 10 + \sqrt[3]{4 + 4} = 48 - 10 + \sqrt[3]{8} = 48 - 10 + 2 = 40$$

Next, we find the limit of $f(x)$ as $x$ approaches $2$: $$\lim_{x \to 2} f(x) = 40$$

To show that $f(x)$ is continuous at $a = 2$, we check if: $$\lim_{x \to 2} f(x) = f(2)$$ Since both values are equal: $$40 = 40$$ This confirms that $f(x)$ is continuous at $x = 2$.

Final Answer