Problem 18
Question
, find the indicated limit. In most cases, it will be wise to do some algebra first. $$ \lim _{h \rightarrow 0} \frac{(x+h)^{2}-x^{2}}{h} $$
Step-by-Step Solution
Verified Answer
The limit is \(2x\).
1Step 1: Recognize the Limit Form
We need to evaluate \( \lim_{h \rightarrow 0} \frac{(x+h)^2 - x^2}{h} \). This expression is in the form of a difference quotient, which commonly appears in calculus as the derivative of a function. We will simplify this expression algebraically before taking the limit.
2Step 2: Expand the Binomial Expression
Expand \((x+h)^2\) using the formula for squaring a binomial: \((a+b)^2 = a^2 + 2ab + b^2\). Thus, \((x+h)^2 = x^2 + 2xh + h^2\).
3Step 3: Simplify the Difference Quotient
Substitute the expanded form into the original expression: \(\frac{(x+h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h}\). Simplify the numerator by canceling the \(x^2\) terms, resulting in \(\frac{2xh + h^2}{h}\).
4Step 4: Factor and Simplify
Factor out \(h\) from the numerator: \(\frac{h(2x + h)}{h}\). Since \(h\) is not zero, we can cancel \(h\) from the numerator and the denominator, resulting in \(2x + h\).
5Step 5: Evaluate the Limit
Now evaluate \(\lim_{h \rightarrow 0} (2x + h)\). As \(h\) approaches zero, the expression simplifies to \(2x\). Therefore, the limit is \(2x\).
Key Concepts
LimitsDifference QuotientBinomial Expansion
Limits
In calculus, limits are fundamental to understanding the behavior of functions as they approach specific points. The concept helps us understand what happens to a function's value as its input gets closer and closer to a particular number. Limits are crucial when we want to find tangents to curves, calculate areas under curves, and understand the behavior of functions near points of discontinuity.
The notation \( \lim_{h \to 0} \) is read as "the limit as \( h \) approaches zero." When solving limit problems, especially those that seem difficult, a little algebra often helps to simplify the process. A limit is about approaching a specific value, not necessarily reaching it. Thus, even if \( h \) cannot actually become zero, we can find how the function behaves in this vicinity and determine what value it tends towards. In our exercise, simplifying the expression using algebra and then applying the limit gives us a clear result.
The notation \( \lim_{h \to 0} \) is read as "the limit as \( h \) approaches zero." When solving limit problems, especially those that seem difficult, a little algebra often helps to simplify the process. A limit is about approaching a specific value, not necessarily reaching it. Thus, even if \( h \) cannot actually become zero, we can find how the function behaves in this vicinity and determine what value it tends towards. In our exercise, simplifying the expression using algebra and then applying the limit gives us a clear result.
Difference Quotient
The difference quotient is a method of calculating the slope of the secant line between two points on a function. It forms the backbone of understanding derivatives in calculus. The basic formula of the difference quotient is \( \frac{f(x+h) - f(x)}{h} \). This expression calculates the average rate of change of the function \( f(x) \) over the interval \( h \) from \( x \) to \( x+h \).
In our original exercise, we use the difference quotient to derive the expression that leads us to determine the function's derivative. Simplifying this expression through algebra (like expansion and factoring) is often essential to evaluate the limit. When \( h \) approaches zero, the difference quotient converges to the derivative, which physically represents an instantaneous rate of change or the slope of the tangent line to the curve at a specific point.
In our original exercise, we use the difference quotient to derive the expression that leads us to determine the function's derivative. Simplifying this expression through algebra (like expansion and factoring) is often essential to evaluate the limit. When \( h \) approaches zero, the difference quotient converges to the derivative, which physically represents an instantaneous rate of change or the slope of the tangent line to the curve at a specific point.
Binomial Expansion
Binomial expansion is a method used to expand expressions that are raised to a power. Specifically, it involves using the binomial theorem, which provides a formula for expanding powers of binomials. For example, \( (a+b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \). This is a powerful tool for algebraic simplifications and is especially useful in calculus when dealing with polynomials and quadratic expressions.
In the exercise we looked at, the expression \( (x+h)^2 \) was expanded using the formula \( a^2 + 2ab + b^2 \). This expansion helped simplify the difference quotient by canceling out terms and factoring, allowing us to find the derivative. Binomial expansion thus plays a critical role in simplifying expressions, making it easier to handle complex algebraic manipulations commonly encountered in calculus problems.
In the exercise we looked at, the expression \( (x+h)^2 \) was expanded using the formula \( a^2 + 2ab + b^2 \). This expansion helped simplify the difference quotient by canceling out terms and factoring, allowing us to find the derivative. Binomial expansion thus plays a critical role in simplifying expressions, making it easier to handle complex algebraic manipulations commonly encountered in calculus problems.
Other exercises in this chapter
Problem 18
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