Question

Evaluating limits 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

Answer: The value of the limit is $$0$$.

Step-by-step solution

Factor the quadratic expression in the numerator

Factor the numerator's quadratic expression using the difference of squares formula: $$x^2 - 2cx + c^2 = (x - c)(x - c)$$ So, the given limit becomes: $$\lim_{x \rightarrow c} \frac{(x-c)(x-c)}{x-c}$$

Simplify the expression by canceling common factors

We notice that \((x - c)\) is a common factor in both numerator and denominator. We can cancel it out: $$\lim_{x \rightarrow c} \frac{(x-c)(x-c)}{x-c} = \lim_{x \rightarrow c} (x - c)$$

Substitute the value of x

Now that we have simplified the expression, substitute the value of \(x\) with \(c\): $$\lim_{x \rightarrow c} (x - c) = c - c$$ Calculating the expression, we get: $$0$$ Therefore, $$\lim _{x \rightarrow c} \frac{x^{2}-2 c x+c^{2}}{x-c} = 0$$

Key concepts

Difference of Squares

The difference of squares is a mathematical identity that helps to factor certain quadratic expressions. An expression can be considered a difference of squares when it is in the form of \( a^2 - b^2 \). This expression can be rewritten and factored as \( (a-b)(a+b) \).
In the original exercise, the numerator \( x^2 - 2cx + c^2 \) resembles a perfect square trinomial, which behaves similarly. It can indeed be factored as \((x-c)(x-c)\).

• The expression \( x^2 - 2cx + c^2 \) can be rewritten using the identity \((x-c)^2\) because it completes the square.
• This creates a more straightforward factorization method compared to handling each term individually.

Recognizing these patterns, especially the structures that fit into familiar identities like the difference of squares, is crucial in simplifying expressions.

Canceling Common Factors

Once we have factored the expression, canceling common factors becomes a powerful simplification step. In a fraction, if the numerator and the denominator have a common factor, it can be "canceled out." This means it can be removed because it appears in both the top and bottom.
In this exercise:

• The factored expression is \((x-c)(x-c)\) in the numerator.
• The denominator is \(x-c\).
• The term \((x-c)\) appears in both the numerator and denominator, allowing us to cancel it out.

When you cancel \((x-c)\) from both numerator and denominator, the expression simplifies!This step is critical because canceling simplifies the limit evaluation, reducing it to a simpler form that is easier to solve.

Substitution Method

The substitution method is a technique used in calculus to evaluate limits effortlessly once an expression has been simplified. After canceling any common factors in a fraction, the expression often allows for direct substitution of the limit point.
Here's how it works in this problem:

• After simplification, the expression becomes \( \lim_{x \rightarrow c} (x-c) \).
• To solve this, we directly substitute \( x \) with \( c \), because the expression is now straightforward.

Executing this substitution:

• Substitute \( x \) with \( c \), which results in \( (c-c) \).
• This simplifies to zero because \( c - c = 0 \).

Substitution is a helpful method because it takes advantage of simplification, making it fast to find the limit once complexities are removed.