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 x🠂1) 3x²+4x-7 / x-1

Verified step by step guidance

1. Step 1: Recognize that the given limit represents the derivative of a function f(x) at a point x = a. The expression (lim x→1) (3x² + 4x - 7) / (x - 1) suggests that we are dealing with a derivative at x = 1.
2. Step 2: Identify the form of the limit. The expression (3x² + 4x - 7) / (x - 1) is in the form of a difference quotient, which is typically used to find the derivative of a function at a specific point.
3. Step 3: Assume that the function f(x) is a polynomial, since the numerator is a polynomial. A reasonable assumption is that f(x) = 3x² + 4x - 7, which is a quadratic function.
4. Step 4: To find the derivative f'(x), apply the power rule to each term of the polynomial: f'(x) = d/dx [3x²] + d/dx [4x] - d/dx [7].
5. Step 5: Evaluate the derivative at x = 1 to find the slope of the tangent line at that point. Substitute x = 1 into f'(x) to calculate the limit.