Problem 70 - Thomas Calculus

Question

You will find a graphing calculator useful. Let $$h(x)=\left(x^{2}-2 x-3\right) /\left(x^{2}-4 x+3\right)$$ a. Make a table of the values of \(h\) at \(x=2.9,2.99,2.999,\) and so on. Then estimate lim_{x} \rightarrow 3 \(h(x)\) . What estimate do you arrive at if you evaluate \(h\) at \(x=3.1,3.01,3.001, \ldots\) instead? b. Support your conclusions in part (a) by graphing \(h\) near \(x_{0}=3\) and using Zoom and Trace to estimate \(y\) -values on the graph as \(x \rightarrow 3\) . c. Find lim_{x\rightarrow3} \(h(x)\) algebraically.

Step-by-Step Solution

Answer

The limit is 4 as x approaches 3.

Step 1: Calculate h(x) at x-values approaching 3 from the left

Start by inputting the function \( h(x) = \frac{x^2 - 2x - 3}{x^2 - 4x + 3} \) into a graphing calculator. Evaluate \( h(x) \) for \( x = 2.9, 2.99, \) and \( 2.999 \). Round the results to a reasonable level of precision. You might find values approaching \( 0.2 \). Continue calculating values until a pattern emerges or singular behavior (like undefined values) is observed.

Step 2: Calculate h(x) at x-values approaching 3 from the right

Use the same function \( h(x) \). This time, evaluate \( h(x) \) for \( x = 3.1, 3.01, \) and \( 3.001 \). Observe the results: Are they close to 0.2 like the values approaching from below? If they present similar behavior, document the pattern or notice any jumps or discontinuities.

Step 3: Graph h(x) around x=3

Graph the function \( h(x) \) using the calculator to gain a visual understanding of its behavior as \( x \) approaches 3. Use the "Zoom" feature to focus on the area around \( x = 3 \). Then, use "Trace" to see how the \( y \)-value changes around \( x = 3 \) to further confirm your numerical and graphical findings.

Step 4: Algebraically simplify and find the limit

To algebraically find \( \lim_{x \to 3} h(x) \), start by factoring the numerator and the denominator: \( x^2 - 2x - 3 = (x-3)(x+1) \) and \( x^2 - 4x + 3 = (x-1)(x-3) \). Notice that \( x-3 \) cancels out, leading to \( h(x) = \frac{x+1}{x-1} \) for \( x eq 3 \). Take the limit \( \lim_{x \to 3} \frac{x+1}{x-1} = 4 \).

Key Concepts

Graphing Calculator

A graphing calculator is a versatile tool that can greatly aid in understanding complex calculus concepts. It allows you to input equations and instantly generate a visual graph of the function, providing insight into its behavior. You can use features like "Zoom" to focus on specific areas, such as around particular x-values where interesting behavior might occur, like approaching a limit.When evaluating functions like \( h(x) = \frac{x^2 - 2x - 3}{x^2 - 4x + 3} \) near a point of interest (in this case, \( x = 3 \)), a graphing calculator can visualize how the function behaves as it approaches this point. This visual analysis, combined with numerical evaluations using specific x-values, helps confirm the behavior you suspect exists based on calculations alone. Using the "Trace" function allows you to follow the curve of the graph closely and see how the y-value changes as x gets closer to the point of discontinuity or interest. This hands-on approach is invaluable for developing a strong intuitive grasp of limits and behavior of rational functions.

Rational Function

Rational functions are fractions where both the numerator and the denominator are polynomials. An example is the function given:\( h(x) = \frac{x^2 - 2x - 3}{x^2 - 4x + 3} \).Rational functions can take on a wide variety of shapes depending on the degree and coefficients of the polynomials involved. One key feature of rational functions is the presence of potential discontinuities, which occur where the denominator is zero. These points do not belong to the function's domain, potentially resulting in vertical asymptotes or removable discontinuities if the factor causing the zero can be canceled.Understanding how rational functions behave around these discontinuities is crucial for graphing and analyzing their behavior near such points. In calculus, examining these areas, like around \( x = 3 \) in our function, is important for estimating limits and understanding the behavior of the function.

Factoring Polynomials

Factoring polynomials is a mathematical skill that simplifies expressions and solves equations, especially useful for rational functions. In the original function:\( x^2 - 2x - 3 \)and \( x^2 - 4x + 3 \),the polynomials can be factored:\( x^2 - 2x - 3 = (x-3)(x+1) \)and\( x^2 - 4x + 3 = (x-1)(x-3) \).By factoring these expressions, you can cancel out common terms in the numerator and the denominator, specifically \( (x-3) \) here, leading to a simplified form of the function for all \( x eq 3 \). This simplification helps in finding the limit of the function as \( x \to 3 \), since the original rational function form might provide undefined results directly due to division by zero.

Continuity and Discontinuity

Continuity in a function means there are no abrupt jumps or gaps in its graph. For rational functions like \( h(x) = \frac{x^2 - 2x - 3}{x^2 - 4x + 3} \),discontinuities often arise when the denominator polynomial equals zero, causing the function to be undefined. In our case, this happens at \( x=3 \) because both the numerator and denominator have a common factor \((x-3)\).Such discontinuities might be removable (a hole in the graph) if the offending factor can be canceled, as it can be here. Otherwise, you might find vertical asymptotes where the function shoots off to infinity.Recognizing and analyzing these points of discontinuity is essential for understanding the overall behavior of the function, particularly in limit calculations. It aids in predicting how the function behaves as it approaches particular x-values from either side.

Graph Analysis

Graph analysis involves visually examining functions to discern key features and behavior. When we graph \( h(x) = \frac{x^2 - 2x - 3}{x^2 - 4x + 3} \),we notice several critical aspects:- **Discontinuities**: Occurrences at points where the function is not defined, such as \( x = 3 \).- **Asymptotes**: Which often occur at undefined points if not canceled, showing where the function diverges.- **Behavior near Points of Interest**: Use "Zoom" to closely examine neighborhood behavior near \( x=3 \), and "Trace" to follow the function's path, providing insights into its limit behavior. Through graph analysis, combined with numeric table values, understanding the left-hand and right-hand limits becomes intuitive. The visual plot complements algebraic simplified forms to verify the correct approaches and interpretations.