Question

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

Short answer

Question: Evaluate the limit of the expression \(\frac{x^2 - 2cx + c^2}{x-c}\) as x approaches the constant c. Answer: The limit of the given expression as \(x\) approaches \(c\) is equal to \(0\).

Step-by-step solution

Factor the numerator

We will begin by factoring the numerator of the given expression: \({x^2 - 2cx + c^2}\). This can be factored as \({(x-c)^2}\). So the expression becomes: \(\frac{(x-c)^2}{x-c}\)

Simplify the expression

Next, we can simplify the expression by canceling out a common factor of \((x-c)\) between the numerator and the denominator. \(\frac{(x-c)^2}{x-c} = \frac{(x-c)(x-c)}{x-c}\) After canceling out \((x-c)\), we get: \(x-c\)

Evaluate the limit

Now, we can evaluate the limit as x approaches the constant c. \(\lim_{x \rightarrow c} (x-c)\) As \(x\) approaches \(c\), the difference \((x-c)\) approaches \(0\). Thus, the limit is: \(\lim_{x \rightarrow c} (x-c) = 0\) In conclusion, the limit of the given expression as \(x\) approaches \(c\) is equal to \(0\).

Key concepts

Evaluating Limits

Understanding how to evaluate limits is fundamental in calculus as it allows us to determine the behavior of a function as it approaches a specific point. The evaluation of limits, particularly when the function is at risk of division by zero, often involves algebraic manipulation to simplify the expression and remove the indeterminate form.

To evaluate a limit, like \(\lim _{x \rightarrow c} \frac{x^{2}-2 c x+c^{2}}{x-c}\), you can't just plug in the value of c directly if it results in an undefined expression. Instead, you apply strategies like factoring polynomials, canceling common terms, and applying limit properties to simplify the function. This process transforms the indeterminate form into one where the limit can be directly computed.

For instance, if the numerator of a fraction is a polynomial that can be factored and shares a common factor with the denominator, this factor can typically be canceled out. After simplification, you might be left with a polynomial or function that can be resolved by direct substitution of the limiting value. This step-by-step approach can turn a perplexing problem into one with a clear solution.

Factoring Polynomials

Factoring polynomials is a vital skill in algebra that comes in handy when evaluating limits in calculus. A polynomial is an expression made up of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents. When factoring, you're breaking down the polynomial into simpler terms (factors) that, when multiplied together, give back the original polynomial.

The given exercise \(\lim _{x \rightarrow c} \frac{x^{2}-2 c x+c^{2}}{x-c}\) starts with a quadratic polynomial in the numerator. Recognizing that this quadratic is a perfect square trinomial—an expression of the form \(a^2 - 2ab + b^2\)—is crucial since it can be factored into \(a - b)^2\). Indeed, we can factor \(x^2 - 2cx + c^2\) into \(x - c)^2\).

This revelation allows us to cancel out the common \(x-c\) factor with the denominator, moving the expression away from an indeterminate form. Mastering polynomial factorization enables students to tackle complex limits by revealing hidden simplifications.

Limit Properties

Limit properties are rules that can help simplify the process of evaluating limits and allow you to do so systematically. These properties are the backbone of limit operations, providing a set of tools to compute the limits of more complex expressions based on the behavior of simpler ones.

Key limit properties include the sum, difference, product, and quotient rules, which allow you to split the limit into smaller parts, provided the limits of those parts exist. These properties hold only if the individual limits are defined. For example, the quotient property lets you divide the limits of two functions if the limit of the denominator is not zero.

In our specific example, after canceling the common term \(x-c\), the property of direct substitution (assuming the function is continuous at the point) can be used to evaluate the limit of the remaining term, which now poses no challenges. Direct substitution, in this case, tells us that as \(x\) approaches \(c\), the expression \(x-c\) approaches zero. Limit properties are the tools that allow students to confidently tackle and simplify limits, ultimately leading to the correct evaluation of the given limit.