Find the function The following limits represent the slope of a curve y = f(x) at the point (a,f(a)). Determine a possible function f and number a; then calculate the limit.
(lim h🠂0) (2+h)⁴-16 / h

Verified step by step guidance

1. Step 1: Recognize that the given limit expression represents the derivative of a function at a point. The expression \( \lim_{h \to 0} \frac{(2+h)^4 - 16}{h} \) is in the form of the difference quotient \( \frac{f(a+h) - f(a)}{h} \), which is used to find the derivative of a function \( f(x) \) at \( x = a \).
2. Step 2: Identify the function \( f(x) \) and the point \( a \). Notice that \( (2+h)^4 \) suggests that \( f(x) = x^4 \) and \( a = 2 \) because \( f(2) = 2^4 = 16 \).
3. Step 3: Confirm that the expression matches the derivative form. Substitute \( f(x) = x^4 \) and \( a = 2 \) into the difference quotient: \( \frac{(2+h)^4 - 2^4}{h} \). This matches the given limit expression.
4. Step 4: Expand \( (2+h)^4 \) using the binomial theorem or direct expansion: \( (2+h)^4 = 16 + 32h + 24h^2 + 8h^3 + h^4 \).
5. Step 5: Substitute the expanded form back into the limit expression: \( \lim_{h \to 0} \frac{32h + 24h^2 + 8h^3 + h^4}{h} \). Simplify by canceling \( h \) from the numerator and denominator, then evaluate the limit as \( h \to 0 \).