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