Finding Limits
In Exercises 9–24, find the limit or explain why it does not exist.




lim h →0 ((x + h)² ― x²)/h

Verified step by step guidance

1. Recognize that the given expression is a difference quotient, which is often used to find the derivative of a function at a point. Here, the function is f(x) = x².
2. Substitute the expression (x + h)² for f(x + h) in the difference quotient: ((x + h)² - x²)/h.
3. Expand the expression (x + h)² using the binomial theorem: (x + h)² = x² + 2xh + h².
4. Substitute the expanded form back into the difference quotient: ((x² + 2xh + h²) - x²)/h.
5. Simplify the expression by canceling out x² and then dividing each term by h: (2xh + h²)/h = 2x + h. Now, take the limit as h approaches 0, which results in the derivative of x² at x.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for defining derivatives and integrals. In this case, we are interested in the limit as h approaches 0, which is essential for evaluating the expression given.

Difference Quotient

The difference quotient is a formula that represents the average rate of change of a function over an interval. It is expressed as (f(x + h) - f(x)) / h, and is used to derive the derivative of a function. In the given limit problem, the expression ((x + h)² - x²)/h is a difference quotient that simplifies to find the derivative of the function f(x) = x².

Derivative

The derivative of a function at a point measures the instantaneous rate of change of the function at that point. It is defined as the limit of the difference quotient as the interval approaches zero. In this exercise, finding the limit will ultimately yield the derivative of the function f(x) = x², which is a key application of limits in calculus.