Question

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

Short answer

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

Step-by-step solution

Expand the expression

Expand the expression inside the limit: \((5+h)^2 = 25 + 10h + h^2\). Now, substitute this back into the expression: $$\lim_{h \rightarrow 0} \frac{(25 + 10h + h^2) - 25}{h}$$

Simplify the expression

Cancel out the common terms in the numerator and denominator, and simplify the expression: $$\lim_{h \rightarrow 0} \frac{10h + h^2}{h} = \lim_{h \rightarrow 0} 10 + h$$

Evaluate the limit

Now, apply the limit by letting \(h\) approach \(0\): $$\lim_{h \rightarrow 0} 10 + h = 10 + 0 = 10$$ So the limit of the given expression as \(h\) approaches \(0\) is \(10\).

Key concepts

Limits

Limits are a fundamental concept in calculus, allowing us to understand the behavior of a function as it approaches a certain point. In simpler terms, a limit helps us find what value a function gets closer to as the input moves towards a particular number. Think of limits as the destination a function is trying to reach.

When dealing with a limit, we denote it with a special notation like \(\lim_{h \rightarrow 0} f(h)\), where \(h\) is the variable that gets closer to zero. The expression inside the limit tells us how the function behaves.

• If the function smoothly approaches a specific value as the variable changes, that specific value is the limit.
• Limits help handle cases where a function might not be well-defined, like division by zero.

Understanding limits is crucial because they form the basis for defining derivatives and integrals, which are the core ideas of calculus.

Limit Evaluation

Evaluating limits means finding what value a function approaches as the variable gets close to a particular point. This requires both reasoning and mathematical manipulation. Here are the steps to evaluate most limits:

• **Direct Substitution**: Try plugging the value directly into the function. If it works without generating undefined forms like \(\frac{0}{0}\), you’ve found the limit.
• **Algebraic Simplification**: If direct substitution fails, simplify the expression using algebra. Factor, expand, or cancel terms to make substitution possible.
• **L'Hôpital's Rule or Other Methods**: For more complex functions, or if simplification doesn't resolve the form, use specific calculus techniques like L'Hôpital's Rule.

In the example given, the limit \(\lim _{h \rightarrow 0} \frac{(5+h)^{2}-25}{h}\) was approached using simplification, allowing the division by \(h\) to not lead to an undefined form.

Algebraic Manipulation

When evaluating limits, algebraic manipulation helps to simplify expressions and avoid indeterminate forms like \(\frac{0}{0}\). In this context, manipulation refers to using fundamental algebraic properties and operations to restructure expressions.

In the given example, algebraic manipulation was used to:

• Expand \((5+h)^2\) to \(25 + 10h + h^2\), showing its full expression.
• Cancel out like terms (here, the constant 25), simplifying the limit to just terms involving \(h\).
• Factor or reduce expressions to help with limit evaluation, as demonstrated by simplifying to \(10 + h\).

These steps aim to transform the expression into one where direct substitution of the limit point becomes possible, making computation straightforward.

Polynomials

Polynomials are algebraic expressions consisting of terms with variables raised to whole number powers. They can take a form such as \(a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\), where \(a_i\) are coefficients.

Polynomials play a crucial role in many areas of mathematics due to their simple structure which makes them easier to manipulate. In calculating limits, dealing with polynomials can often simplify tasks:

• They have predictable behavior as inputs change, making limits straightforward.
• Each term can be handled individually using algebraic rules.
• Polynomial expressions do not trap common pitfalls like division by zero unless manipulated improperly.

In our exercise, manipulating the polynomial expression \((5+h)^2\) was essential to simplify the process of limit evaluation. This showcases why understanding polynomials can streamline complex calculus operations.