Question

Finding a Limit In Exercises \(41-46,\) write a simpler function that agrees with the given function at all but one point. Then find the limit of the function. Use a graphing utility to confirm your result. $$ \lim _{x \rightarrow-2} \frac{3 x^{2}+5 x-2}{x+2} $$

Short answer

The limit of the given function as \(x\) approaches -2 is -7.

Step-by-step solution

Factorize the function

First, factorize the numerator of the function to see if it can be simplified. Since \(3x^2 + 5x - 2\) can be factored into \((3x -1)(x+2)\), we get \(\frac{(3x -1)(x+2)}{x+2}\).

Simplify the function

Recognize that the \(x+2\) term in the numerator and denominator cancel each other out, leaving just \(3x - 1\), which is valid for all \(x\) not equal to -2.

Find the limit

We can now find the limit as \(x\) approaches -2 for this new function \(3x -1\). Plugging -2 into the function yields \(-3*2 -1 = -7\).

Confirm the result using a graphing utility

Graph both the initial function and the new function to see that they give the same output for all \(x\) except when \(x = -2\). When \(x = -2\), confirm as well that the new function results in -7, which validates your previous result.

Key concepts

Simplifying Rational Functions

Simplifying rational functions is a crucial first step in evaluating limits, especially when the function involves a removable discontinuity. A rational function is typically characterized by a polynomial in the numerator and another in the denominator. If these polynomials share a common factor, simplification can be achieved by canceling out these common factors.

In this example, the rational function given is \(\frac{3x^2 + 5x - 2}{x + 2}\). This represents a typical case where simplification is needed before evaluating the limit. By simplifying the function, complications such as undefined limits or misleading values at specific points can be avoided.

Simplifying will make it possible to consider the behavior of the function around troublesome points without being concerned about division by zero.

Factoring Polynomials

Factoring polynomials is an essential technique in simplifying rational functions. It allows us to express a polynomial as a product of its linear factors, which can then be canceled if shared with the denominator.

For the polynomial \(3x^2 + 5x - 2\) in the numerator, we need to find two numbers that multiply to \(-6\) (from \(3 \times -2\)) and add to \(5\). By using factorization, we find that \(3x^2 + 5x - 2 = (3x - 1)(x + 2)\).

This step is vital in breaking down complex expressions into more manageable segments. It prepares the function for further simplification by highlighting potential common factors with the denominator. Once factored, these parts of the polynomial can often be canceled, facilitating a clearer path to finding the limit."

Canceling Common Factors

After factoring the polynomial, the next step in simplifying a rational function is canceling common factors. This step is key in transforming the function into a simpler equivalent form, reducing the complexity involved when evaluating limits.

In our function, \((3x - 1)(x + 2)\) is over \(x + 2\). Here, \(x + 2\) appears in both the numerator and the denominator. Canceling this common factor cancels out what would otherwise be an undefined point at \(x = -2\), and simplifies our function to just \(3x - 1\).

Be mindful that this cancellation removes the specific restriction in the domain at \(x = -2\), making the simpler form valid everywhere else. The removal of common factors simplifies limits evaluation and serves to highlight the function's behavior as it approaches particular points.

Evaluating Limits Analytically

Once a function has been simplified by factoring and canceling common terms, it becomes easier to evaluate the limit. Evaluating limits analytically involves directly substituting the point into the simplified function, provided there are no discontinuities at that point.

For the simplified function \(3x - 1\), to find \(\lim_{x \rightarrow -2} (3x - 1)\), we substitute \(x = -2\) into our expression, which gives us \(-3\times -2 - 1 = -7\).

This analytical approach provides a straightforward means to identify how the function behaves near the point in question. The result is clearer due to the elimination of undefined expressions present before simplification. By using a graphing utility, students can also visually confirm the accuracy of this limit across other x-values, ensuring a comprehensive understanding of the function’s behavior.