Problem 23 - Calculus Early Transcendentals

Question

Determining limits analytically Determine the following limits or state that they do not exist. a. \(\lim _{x \rightarrow 2^{+}} \frac{x^{2}-4 x+3}{(x-2)^{2}}\) b. \(\lim _{x \rightarrow 2^{-}} \frac{x^{2}-4 x+3}{(x-2)^{2}}\) c. \(\lim _{x \rightarrow 2} \frac{x^{2}-4 x+3}{(x-2)^{2}}\)

Step-by-Step Solution

Answer

Answer: The limits are as follows: a. Right-hand limit: \(\lim _{x \rightarrow 2^{+}} f(x) = -1\) b. Left-hand limit: \(\lim _{x \rightarrow 2^{-}} f(x) = -1\) c. Overall limit: \(\lim _{x \rightarrow 2} f(x) = -1\)

Step 1: Observing and simplifying the function

Observe the function: \(\frac{x^2 - 4x + 3}{(x-2)^2}\). First, let's simplify the numerator: \(x^2 - 4x + 3 = (x-1)(x-3)\). So the function becomes: \(f(x) = \frac{(x-1)(x-3)}{(x-2)^2}\).

Step 2: Determining the right-hand limit

To determine the right-hand limit, we need to analyze the function as \(x \rightarrow 2^+\). For \(x > 2\), \((x-1), (x-2)\) and \((x-3)\) are positive. Thus, the function will be positive as \(x \rightarrow 2^+\). Now let's find the limit: $$\lim _{x \rightarrow 2^{+}} \frac{(x-1)(x-3)}{(x-2)^{2}}$$. As \(x \rightarrow 2\), we have: $$(2^+ - 1)(2^+ - 3) \rightarrow 1 \cdot (-1) = -1$$. Thus, $$\lim _{x \rightarrow 2^{+}} \frac{(x-1)(x-3)}{(x-2)^{2}} = -1$$.

Step 3: Determining the left-hand limit

To determine the left-hand limit, we need to analyze the function as \(x \rightarrow 2^-\). For \(x<2\), \((x-1)\) is negative, while \((x-2)\) and \((x-3)\) are positive. This makes the function positive as \(x \rightarrow 2^-\). Now let's find the limit: $$\lim _{x \rightarrow 2^{-}} \frac{(x-1)(x-3)}{(x-2)^{2}}$$. As \(x \rightarrow 2\), we have: $$(2^- - 1)(2^- - 3) \rightarrow 1\cdot (-1) = -1$$. Thus, $$\lim _{x \rightarrow 2^{-}} \frac{(x-1)(x-3)}{(x-2)^{2}} = -1$$.

Step 4: Determining the overall limit of the function

We found the right-hand and left-hand limit of the function as x approaches 2. Both of them have the same value, which means the overall limit of the function exists and is equal to the common value: $$\lim _{x \rightarrow 2} \frac{(x-1)(x-3)}{(x-2)^{2}} = -1$$. Final answer: a. \(\lim _{x \rightarrow 2^{+}} \frac{x^{2}-4 x+3}{(x-2)^{2}} = -1\) b. \(\lim _{x \rightarrow 2^{-}} \frac{x^{2}-4 x+3}{(x-2)^{2}} = -1\) c. \(\lim _{x \rightarrow 2} \frac{x^{2}-4 x+3}{(x-2)^{2}} = -1\)

Key Concepts

Right-Hand Limit

When evaluating the right-hand limit, we focus on the behavior of a function as the input approaches a particular value from the right side. In mathematical terms, it involves approaching the value from numbers that are slightly greater than the target. This concept is crucial when continuity or different behaviors are present on each side of the point.

For instance, consider the limit \( \lim _{x \rightarrow 2^{+}} \frac{(x-1)(x-3)}{(x-2)^{2}} \). Here, \(x \rightarrow 2^{+}\) means values just a bit more than 2. As we simplify, for any \( x > 2 \), expressions \((x-1), (x-2),\) and \((x-3)\) are all greater than zero, except when approached from the positive side, the expressions retain only their absolute values magnified.

Thus, we see that

• \((x-1) \) is positive,
• \((x-2)^2 \) is very small (denominator approaches zero),
• So, the large positive numerator divided by a small positive denominator approaches infinity or similar large values.

Understanding this helps in predicting the behavior of the function or sequence from the specific side and identifies points of potential discontinuity.

Left-Hand Limit

The left-hand limit assesses the function's behavior as the variable approaches a specific number from the left, i.e., from numbers slightly less than the target value. This is particularly important when examining functions that may behave differently on each side of the point.

In the expression \( \lim _{x \rightarrow 2^{-}} \frac{(x-1)(x-3)}{(x-2)^{2}} \), the notation \(x \rightarrow 2^{-}\) indicates we examine values slightly less than 2. As such, let's observe:

• For \( x<2 \), the numerator \((x-1)\) is less than zero,
• \((x-2)\) is still small (approaching zero), making \((x-2)^2\) also small but positive, and
• \((x-3)\) remains positive.

Hence, the overall fraction approaches a negative value on this side.

Recognizing left-hand limits can be vital in recognizing an asymmetry in function behavior, and they're a building block for determining broader conclusions about the function across intervals.

Analytical Limit Determination

Analytical limit determination is a method of using algebraic simplification and logical deduction to find the limit of a function as it approaches a specific value. It helps identify how a function behaves locally around a point, without necessarily graphing it.

In the example we tackled: \( \lim _{x \rightarrow 2} \frac{(x-1)(x-3)}{(x-2)^{2}} \), we firstly expanded and factored expressions into simpler forms. This simplification exposed potentially removable discontinuities or behavior trends in the function.

What follows includes:

• observing how factors become dominant when \( x \) nears specific values,
• focusing on zero denominators or other undefined expressions to decide the limit's direction,
• and finally, matching both one-sided limits together to derive the overall limit.

Analytical methods often give more immediate insights into continuity, possible infinite limits, and overall function stability behavior at critical points.