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.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, the limit helps determine the slope of the curve at a specific point, which is essential for understanding the behavior of the function near that point.

Derivatives

The derivative of a function at a point gives the slope of the tangent line to the curve at that point. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In this problem, calculating the limit will help us find the derivative of the function at the specified point.

Function Evaluation

Function evaluation involves substituting a specific value into a function to find its output. In this case, we need to determine a function f and a point a, then evaluate the limit to find the slope at that point. This process is crucial for connecting the limit to the actual behavior of the function.