Problem 5 - Precalculus Mathematics for Calculus

Question

Evaluate the limit and justify each step by indicating the appropriate Limit Law(s). $$\lim _{x \rightarrow 4}\left(5 x^{2}-2 x+3\right)$$

Step-by-Step Solution

Answer

The limit is 75.

Step 1: Identify the Limit Expression

We need to find the limit of the polynomial function \(5x^2 - 2x + 3\) as \(x\) approaches 4. This is a limit of a polynomial, which is a straightforward application of limit laws.

Step 2: Apply the Limit to Each Term

According to the Sum/Difference Rule for limits, we can apply the limit individually to each term of the polynomial. Thus: \(\lim_{x \rightarrow 4} (5x^2 - 2x + 3) = \lim_{x \rightarrow 4} (5x^2) - \lim_{x \rightarrow 4} (2x) + \lim_{x \rightarrow 4} (3)\).

Step 3: Use the Constant Multiple and Power Rule

Apply the Constant Multiple Rule (\(c \cdot f(x)\) becomes \(c \cdot \lim_{x \rightarrow a} f(x)\)) and the Power Rule (\((x^n)\) becomes \((a^n)\)). Thus: 1. \(\lim_{x \rightarrow 4} (5x^2) = 5\lim_{x \rightarrow 4} (x^2) = 5(4^2) = 80\). 2. \(\lim_{x \rightarrow 4} (2x) = 2\lim_{x \rightarrow 4} (x) = 2(4) = 8\). 3. \(\lim_{x \rightarrow 4} 3 = 3\).

Step 4: Combine the Results

Add the results of each term's limit: \(\lim_{x \rightarrow 4} (5x^2 - 2x + 3) = 80 - 8 + 3\).

Step 5: Simplify the Expression

Simplify the final expression: \(80 - 8 + 3 = 75\). Thus, the limit of the expression as \(x\) approaches 4 is 75.

Key Concepts

Limit Laws

Limit laws are rules that help us evaluate limits of functions in a systematic way. They are like stepping stones that guide us through calculating complex limits. Some core limit laws include:

• Sum/Difference Rule: This rule states that the limit of a sum/difference is the sum/difference of the limits. So, for a function like \(f(x) + g(x)\), the limit is \(\lim_{x \to a} f(x) + \lim_{x \to a} g(x)\).


• Constant Multiple Rule: With this rule, if you have a constant \(c\) multiplying a function \(f(x)\), the limit is \(c \times \lim_{x \to a} f(x)\).


• Power Rule: For a function with \(x^n\), the limit can be evaluated as \((\lim_{x \to a} x)^n = a^n\).

These laws help break down complex functions into simpler parts to make limit calculations manageable and understandable.

Polynomial Functions

Polynomial functions are expressions composed of variables raised to whole number powers and their coefficients. Simply put, they are like a sequence of terms involving powers of \(x\). One great thing about polynomials is how easy they are to handle when computing limits.

• Each term like \(ax^n\) can be separately evaluated by standard limit laws and combined using the Sum/Difference Rule.


• In general, evaluating the limit of a polynomial as \(x\) approaches some number \(a\) is quite straightforward: Plug \(a\) directly into the polynomial.

For instance, with a polynomial function like \(5x^2 - 2x + 3\), you can individually compute the limit for each term and sum those results, making the calculations very efficient.

Sum/Difference Rule

The Sum/Difference Rule is a basic yet powerful tool when working with limits. It states that one can take the limit of each individual term, whether you're adding or subtracting them in the expression.

• For example, if you have terms \(f(x)\) and \(g(x)\), calculate \(\lim_{x \to a} f(x)\) and \(\lim_{x \to a} g(x)\) separately, then find \(\lim_{x \to a} (f(x) \pm g(x))\).


• This rule allows us to split complex expressions into manageable calculations.

Consider the function \(5x^2 - 2x + 3\); by applying this rule, we calculate the limit for \(5x^2\), \(-2x\), and \(+3\), combining their respective limits to find the overall limit.

Constant Multiple and Power Rule

The Constant Multiple and Power Rules are essential for simplifying limit calculations involving polynomial functions.

• Constant Multiple Rule: When a function is multiplied by a constant, like \(cf(x)\), we can take \(c\) out and multiply it with the limit of the function. So \(\lim_{x \to a} cf(x) = c \times \lim_{x \to a} f(x)\). This rule makes it easy when working with terms like \(5x^2\) or \(-2x\).


• Power Rule: With powers, the rule tells us \(\lim_{x \to a} (x^n) = a^n\). So when a term like \(x^2\) is present, just replace \(x\) with \(a\) and compute the power.

These rules simplify handling explicit powers and constants in limit calculations by breaking them down logically, making them essential tools, especially in the study of calculus.