Question

Determine if \(f(x)=\left\\{\begin{array}{ll}\frac{x^{2}+5 x-14}{x-2}, & \text { if } x \neq 2 \\ 12, & \text { if } x=2\end{array}\right.\) continuous at \(x=2\). Explain why or why not.

Short answer

Answer: No, the function is not continuous at \(x=2\), as the limit of the function as \(x\) approaches 2 (which is 9) is not equal to the function value at \(x=2\) (which is 12).

Step-by-step solution

Check if the function is defined at x=2

The function is defined as following for x=2: \(f(2) = 12\) So it's defined at x=2.

Calculate the limit as x approaches 2

Find the limit of the function as x approaches 2. For this piecewise function, we will focus on the first part of the function, which is defined for all x not equal to 2: \(\lim_{x\to 2} \frac{x^2 + 5x - 14}{x - 2}\)

Simplify the expression inside the limit

To evaluate this limit, we first factor the numerator: \(x^2 + 5x - 14 = (x - 2)(x + 7)\) Now we have: \(\lim_{x\to 2} \frac{(x - 2)(x + 7)}{x - 2}\)

Cancel the common factor

We can cancel the \((x - 2)\) term from both the numerator and the denominator: \(\lim_{x\to 2} x + 7\)

Evaluate the limit

Now we can find the limit by plugging in the value of \(2\): \(\lim_{x\to 2} (x + 7) = 2 + 7 = 9\)

Compare the limit with the function value at x=2

The limit value as x approaches 2 is \(9\). However, the function value at x=2 is \(12\). Since the limit of the function as x approaches 2 is not equal to the function value at x=2, we can conclude that the given function is not continuous at x=2.

Key concepts

Piecewise Functions

A piecewise function is one that is defined by different expressions for different intervals of the variable. In our exercise, the function \(f(x)\) is given as a piecewise function with two different rules: one that applies when \(x\) is not equal to 2, and another when \(x\) is exactly 2.

Understanding how to work with these functions is essential for calculus students, especially when evaluating limits and determining continuity. Determining if a piecewise function is continuous at a point involves checking if the function value at that point is defined and whether limit of the function as \(x\) approaches the point from any direction equal the function's value at that point.

When working with piecewise functions, one must also ensure that the function 'connects' properly at the boundaries defined by the piecewise intervals. If there is a 'jump' in the function at any of those points, then it may be discontinuous at that point.

Limits

Limits in calculus are used to describe the value that a function approaches as the input approaches a certain value. In the context of continuity, a limit is crucial for determining whether a function has a jump, hole, or is otherwise not smooth at a particular point.

In our exercise, we need to calculate the limit of \(f(x)\) as \(x\) approaches 2, which means we are interested in the function's behavior around but not at the point x=2. Even if \(f(x)\) is not defined at that specific point, it can be continuous if the limit matches the defined function value at that point.

If evaluating the limit of a piecewise function directly at the point of interest leads to an undefined expression, as it does here with the \((x - 2)\) in the denominator, we look for a way to simplify the function. Otherwise, it would imply the existence of a hole or vertical asymptote, leading to discontinuity.

Factorization in Calculus

Factorization plays a significant role in simplifying expressions and solving equations in calculus. By factoring the numerator or denominator, we can often cancel common terms and make a limit easier to evaluate.

In our problem, the step involving factorization is vital. The numerator factors into \((x - 2)(x + 7)\), which allows us to cancel the \((x - 2)\) term in the denominator. Without this step, we could not evaluate the limit as \(x\) approaches 2, because it would result in division by zero.

After the cancellation, we end up with a simpler expression that can be directly evaluated to determine the limit. But remember, factorization is not just a mechanical step; it provides insights into the function's behavior and reveals potential points of discontinuity that are important for understanding the overall behavior of functions in calculus.