Problem 65 - Calculus Early Transcendentals

Question

Evaluate the following limits, where \(c\) and \(k\) are constants. \(\lim _{h \rightarrow 0} \frac{(5+h)^{2}-25}{h}\)

Step-by-Step Solution

Answer

Answer: The limit of the function as \(h\) approaches \(0\) is \(10\).

Step 1: Expand the numerator

Expand \((5+h)^2\) in the numerator using the binomial formula: \((a+b)^2 = a^2 + 2ab + b^2\) So, \((5+h)^2 = 5^2 + 2(5)(h) + h^2 = 25 + 10h + h^2\).

Step 2: Substitute the expanded numerator

Substitute the expanded numerator back into the original expression: \(\lim_{h\rightarrow 0} \frac{(25+10h+h^2)-25}{h}\)

Step 3: Simplify the expression

Simplify the expression by canceling out the constant terms in the numerator: \(\lim_{h\rightarrow 0} \frac{10h+h^2}{h}\)

Step 4: Factor out \(h\) from the numerator

Factor out \(h\) from the numerator to cancel it with the denominator: \(\lim_{h\rightarrow 0} \frac{h(10+h)}{h}\)

Step 5: Cancel common factors

Cancel the common factors of \(h\) from the numerator and denominator: \(\lim_{h\rightarrow 0} (10+h)\)

Step 6: Evaluate the limit

As \(h\) approaches \(0\), the expression becomes simply \(10\): \(\lim_{h\rightarrow 0} (10+h) = 10\) The final answer is \(10\).

Key Concepts

Binomial Formula

Understanding the binomial formula is key to handling algebraic expressions that involve the sum of two terms raised to a power. In the context of our problem, we use the standard binomial expansion for the square of a sum, \( (a+b)^2 = a^2 + 2ab + b^2 \). This allows us to transform \( (5+h)^2 \) into a more manageable form by recognizing the pattern of \( a=5 \) and \( b=h \). Once we've expanded \( (5+h)^2 \) using this formula, we are left with a straightforward expression \( 25 + 10h + h^2 \) that can be easily simplified further. The binomial formula is instrumental in calculus because it turns complicated limit problems into simpler ones that can be algebraically manipulated.

Simplifying Expressions

Simplifying expressions helps in making complex algebraic equations more understandable and easier to work with. In our given limit problem, simplification involves canceling out constants and like terms. By subtracting \( 25 \) from \( 25+10h+h^2 \), we effectively remove the static part of the expression, leaving us with terms that all contain \( h \) as a factor. Simplification is a crucial step in solving limits, as it often reveals the underlying structure of the function and allows us to proceed without unnecessary complications. It's the process of peeling back the layers of an expression to reveal its simplest form, which is an essential skill for any student tackling calculus.

Factorization

Factorization is a technique used to break down expressions into products of simpler factors, making them easier to manipulate, especially when dealing with limits. After simplifying our expression, we are left with \( 10h+h^2 \) in the numerator. By factoring out the common \( h \) from both terms, the expression becomes \( h(10+h) \). This step is crucial as it sets us up for the next move: canceling out the \( h \) from both the numerator and the denominator, which is only possible because the terms are now in a factored form. Grasping factorization allows students to navigate smoothly through complex algebra and calculus problems, ultimately revealing the behavior of functions as certain variables approach specific values.

Limits of Functions

Finally, the concept of the limits of functions is where calculus truly begins. A limit examines the behavior of a function as it approaches a certain point. After taking the steps to expand, simplify, and factor our function, we arrived at the simplified form \( 10+h \). By taking the limit as \( h \) approaches zero, we determine what value the function is approaching, not necessarily the function's actual value at that point. In this case, the function approaches the value \( 10 \). Understanding limits is fundamental in calculus because they are used to define derivatives, integrals, and the continuity of functions, which are the cornerstones of this mathematical branch. The limit tells us about the behavior of functions 'at the edge'—or, in technical terms, in the immediate vicinity of particular points or as variables grow large.