Question Statement

Determine whether the function $f(x)$ is continuous at $x=6$, where:

$f(x)=\{\begin{array}{ll}\frac{x^2-36}{x-6}& \text{if }x\ne 6\\ 12& \text{if }x=6\end{array}$

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Background and Explanation

A function $f(x)$ is continuous at a point $x=a$ if the limit as $x$ approaches $a$ equals the function value at $a$, i.e., ${\lim }_{x\to a}f(x)=f(a)$. For piecewise functions with rational expressions, we often need to factor and simplify to evaluate the limit at points where the expression is undefined.

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Solution

To determine continuity at $x=6$, we must verify whether ${\lim }_{x\to 6}f(x)=f(6)$.

Step 1: Calculate the limit as $x\to 6$

For $x\ne 6$, the function is defined as $\frac{x^2-36}{x-6}$. We evaluate:

${\lim }_{x\to 6}f(x)={\lim }_{x\to 6}\frac{x^2-36}{x-6}$

Step 2: Factor the numerator

Recognize that $x^2-36$ is a difference of squares, which factors as $(x-6)(x+6)$:

$={\lim }_{x\to 6}\frac{(x-6)(x+6)}{x-6}$

Step 3: Simplify the expression

Since $x\to 6$ but $x\ne 6$ in the limit process, we can cancel the common factor $(x-6)$ from the numerator and denominator:

$={\lim }_{x\to 6}(x+6)$

Step 4: Evaluate the limit

Substitute $x=6$ into the simplified expression:

$=6+6=12$

Therefore: ${\lim }_{x\to 6}f(x)=12$

Step 5: Check the function value

From the piecewise definition, we are given: $f(6)=12$

Step 6: Compare and conclude

Since ${\lim }_{x\to 6}f(x)=12$ and $f(6)=12$, we have: ${\lim }_{x\to 6}f(x)=f(6)$

Therefore, $f(x)$ is continuous at $x=6$.

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Key Formulas or Methods Used

• Condition for continuity: ${\lim }_{x\to a}f(x)=f(a)$
• Difference of squares factorization: $a^2-b^2=(a-b)(a+b)$
• Limit of rational functions: Factor and cancel common terms to resolve indeterminate forms $\frac{0}{0}$

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Summary of Steps

1. Set up the limit: Write ${\lim }_{x\to 6}\frac{x^2-36}{x-6}$ for the $x\ne 6$ case
2. Factor and simplify: Factor the numerator as $(x-6)(x+6)$ and cancel $(x-6)$
3. Evaluate: Calculate ${\lim }_{x\to 6}(x+6)=12$
4. Compare values: Check that $f(6)=12$ matches the limit
5. Conclude: Since ${\lim }_{x\to 6}f(x)=f(6)$, the function is continuous at $x=6$