Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(\lim _{x \rightarrow 2}[f(x)-f(2)] /(x-2)\) for each given function \(f\). $$f(x)=3 x^{2}$$

Step-by-Step Solution

Answer

The limit is 12.

Step 1: Define the Function at the Limit

First, identify the function at the given point. In this case, we have the function \(f(x) = 3x^2\). Calculate \(f(2)\) to start simplifying the expression inside the limit:\[ f(2) = 3(2)^2 = 3 \times 4 = 12 \]

Step 2: Simplify the Expression Inside the Limit

Write the expression \([f(x) - f(2)]\) in the numerator of the limit. This looks like:\[ f(x) - f(2) = 3x^2 - 12 \]

Step 3: Rewrite the Limit Expression

Substitute the simplified expression from Step 2 into the original limit problem:\[ \lim_{x \to 2} \frac{3x^2 - 12}{x-2} \]

Step 4: Factor the Numerator

Factor the expression \(3x^2 - 12\). Notice that this expression can be factored as follows:\[ 3x^2 - 12 = 3(x^2 - 4) = 3(x - 2)(x + 2) \]

Step 5: Cancel Common Factors

Now that we have the numerator factored, we can cancel the common \((x - 2)\) factor from the numerator and the denominator:\[ \frac{3(x - 2)(x + 2)}{x - 2} = 3(x + 2) \] (for \(x eq 2\))

Step 6: Evaluate the Limit

Now that we have canceled the \((x - 2)\) term, evaluate the limit:\[ \lim_{x \to 2} 3(x + 2) = 3(2 + 2) = 3 \times 4 = 12 \]

Key Concepts

Derivative

The derivative of a function at a particular point gives us the rate at which the function's value changes as its input changes. In the given problem, we want to find the derivative of the function using the definition of a derivative. This involves calculating the limit:
\[\lim_{x \to a} \frac{f(x) - f(a)}{x - a}\]at a point where \(x\) approaches a value \(a\). For the quadratic function \(f(x) = 3x^2\), calculating its derivative at \(x = 2\) requires evaluating this specific limit.

• The expression \(f(x) - f(2)\) represents the change in function values.
• Dividing by \(x - 2\) gives the average rate of change over the interval \([x, 2]\).

By finding the limit as \(x\) approaches 2, we arrive at the instantaneous rate of change, which is the derivative of \(f(x)\) at \(x = 2\).

Limit Evaluation

Limit evaluation is crucial for understanding how functions behave as inputs approach a particular value. In our exercise, the limit expression is:
\[\lim_{x \to 2} \frac{f(x) - f(2)}{x - 2}\]which simplifies to the derivative at \(x = 2\). To evaluate limits like this, you can:

• Simplify the expression, often by factoring.
• Cancel common factors where possible.
• Substitute the limit value into the simplified expression.

Sometimes, limits might lead to indeterminate forms like 0/0, which requires manipulation, such as factoring, to resolve. In this example, factoring helps us cancel the troublesome term \((x - 2)\). Once simplified, plug the limiting value into the equation to find the result.

Polynomial Functions

Polynomial functions, like \(f(x) = 3x^2\), are algebraic expressions composed of terms involving powers of a variable. They are characterized by:

• Power terms: \(3x^2\) has a power of 2, signifying a quadratic polynomial.
• Coefficients: The number 3 in \(3x^2\) is the coefficient that scales the term.

Polynomials are common in calculus because they are easy to differentiate and integrate. In our exercise, the polynomial is quadratic, meaning its graph is a parabola. This particular shape allows us to easily apply calculus techniques like differentiation to analyze changes and rates of change.

Factoring Techniques

Factoring is a method used to break down expressions into simpler products of expressions. It's especially useful in calculus when evaluating limits of polynomial functions, since factoring can make it easier to cancel terms and prevent indeterminate forms like 0/0. For the expression \(3x^2 - 12\), the factoring process involves:

• Identifying common factors: The number 3 is a factor of both terms.
• Using algebraic techniques: Recognize the difference of squares \((x^2 - 4)\).
• Resulting factorization: \(3(x - 2)(x + 2)\).

By factoring \(3x^2 - 12\) into \(3(x - 2)(x + 2)\), we can cancel the \((x - 2)\) from the numerator and denominator, simplifying the limit evaluation. This technique is a fundamental tool for handling polynomial limits and derivatives.