Problem 5 - Precalculus: Mathematics for Calculus

Question

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 5} x$$

Step-by-Step Solution

Answer

The limit is 5, using the Identity Law for limits.

Step 1: Identify the Expression

The expression we need to evaluate is \( \lim_{x \rightarrow 5} x \). This expression is simply \( x \) approaching the value of 5 as \( x \) gets closer to 5.

Step 2: Apply the Identity Law

The Identity Law for limits states that if \( f(x) = x \), then \( \lim_{x \rightarrow a} f(x) = a \). In this case, we have \( f(x) = x \) and \( a = 5 \). Therefore, we can directly apply this law and conclude that \( \lim_{x \rightarrow 5} x = 5 \).

Key Concepts

Limit Laws

Limit laws serve as the rulebook for evaluating limits in calculus. They provide a set of guidelines that allow us to simplify and solve limit problems. These laws include:

• The Sum Law: which allows us to take the limit of the sum of two functions by taking the limits of the functions separately and then adding them.
• The Product Law: similar to the sum law but applied to multiplication.
• The Quotient Law: which gives us the limit of a quotient of two functions, as long as the limit of the denominator isn't zero.
• The Constant Multiple Law: can be used when a constant is multiplied by a function, allowing us to pull the constant outside of the limit process.
• The Power Law: useful when dealing with a function raised to an exponent.

Each of these laws simplifies otherwise complex expressions, making limit problems manageable. They form the foundational tools needed to effectively solve most limit problems in calculus.

Identity Law

The Identity Law is one of the simplest yet powerful limit laws. It helps evaluate the limit of a function by providing a direct approach when the function is simply the variable itself, like in our exercise: When applying this law, consider a function such as \( f(x) = x \). The Identity Law states that:\[\lim_{x \rightarrow a} f(x) = a\]This means if \( f(x) \) is just \( x \), as \( x \) approaches a particular value \( a \), the limit of \( f(x) \) is simply \( a \).In simpler terms, if you're asked to evaluate \( \lim_{x \rightarrow 5} x \), you can immediately conclude that the limit is 5. This law relies on the fact that as the variable approaches a point, the value of the variable approaches that same point. This makes the Identity Law incredibly straightforward and useful when evaluating basic limits.

Evaluating Limits

Evaluating limits, in calculus, is an essential skill that involves determining the behavior of a function as the input approaches a given value. Understanding and applying limit laws like the Identity Law, significantly streamlines this process. To evaluate a limit:

• First, identify if any specific limit laws apply to the problem at hand.
• If the function is a simple expression, directly apply the Identity Law.
• In more complex functions, break down the expression using relevant limit laws, such as sum, product, and quotient laws.
• Always watch for points where the function might be undefined, like division by zero.

Here's a tip: if a function appears complex, check if simplifying it can reveal opportunities to apply limit laws. Through practice, recognizing applicable laws and evaluating limits becomes intuitive, ensuring success with calculus problems.