Problem 30 - Calculus: Early Transcendentals

Question

Determine the following limits. a. \(\lim _{x \rightarrow 1^{+}} \frac{x-3}{\sqrt{x^{2}-5 x+4}}\) b. \(\lim _{x \rightarrow 1} \frac{x-3}{\sqrt{x^{2}-5 x+4}}\) c. \(\lim _{x \rightarrow 1} \frac{x-3}{\sqrt{x^{2}-5 x+4}}\)

Step-by-Step Solution

Answer

The results for the three different limits are: a. The limit does not exist as x approaches 1 from the right. b. The limit does not exist as x approaches 1 from both sides. c. The limit does not exist as x approaches 1 from both sides (repeated case).

Step 1: Analyze the function

In all three cases, we are dealing with the function \(g(x) = \frac{x-3}{\sqrt{x^{2}-5 x+4}}\). We can rewrite the denominator as $$\sqrt{x^2 - 5x + 4} = \sqrt{(x - 4)(x - 1)}$$. This factored form of the denominator will help us identify if there are any troublesome areas near x=1.

Step 2: Case a solution - Approaching from the right

For case a, our task is to find \(\lim _{x \rightarrow 1^{+}} \frac{x-3}{\sqrt{x^{2}-5 x+4}}\). To do this, we'll analyze what happens as x gets closer and closer to 1 from the right (positive direction). As \(x \rightarrow 1^{+}\), \((x - 4)\) will always be negative, and \((x - 1)\) will always be positive. This means the denominator will be of the form \(\sqrt{(\text{negative})(\text{positive})}\). Since there is no real number that can be the square root of a negative number, the limit does not exist for this case.

Step 3: Case b solution - Approaching from both sides

For case b, we need to determine the limit as \(x \rightarrow 1\). In other words, we need to make sure the function approaches the same value from both sides. However, we already found that the function does not approach any value from the right side in step 2. In this case, we can automatically conclude that the limit for case b does not exist either.

Step 4: Case c solution - Approaching from both sides (repeated)

Case c is identical to case b. Consequently, the function does not approach any value from both sides; thus, the limit does not exist for case c as well. Here are the answers for all three cases: a. The limit does not exist. b. The limit does not exist. c. The limit does not exist.

Key Concepts

One-sided Limits

In calculus, one-sided limits are used to analyze the behavior of a function as the input approaches a particular point from one side only—either from the left \( (x \rightarrow c^-) \) or the right \( (x \rightarrow c^+)\). This is crucial when the function behaves differently on each side of the point or when the function is not defined on one side of that point.

For example, consider the function \( f(x) = \frac{x-3}{\sqrt{x^2 - 5x + 4}} \) as you approach \( x=1 \) from the right \( (x \rightarrow 1^+) \). The function is not defined at exactly \( x=1 \) and behaves differently immediately to the left and right of \( x=1 \). As \( x \) approaches 1 from the right, the denominator approaches 0 through negative values, which is impossible for a square root of a real number. Therefore, the limit as \( x \) approaches 1 from the right does not exist at all, showcasing the necessity of evaluating one-sided limits.

Limit Does Not Exist

When we say that a limit does not exist, we are acknowledging that as \( x \) approaches a particular value, the function \( f(x) \) does not approach any specific number, approaches different numbers from the left and right, or grows without bound. Not existing doesn't simply mean that the function isn't approaching a real number; rather, it could also mean that there's no single value to approach due to a contradiction or discontinuity.

In the given exercise, the function \( g(x) = \frac{x-3}{\sqrt{x^2 - 5x + 4}} \) cannot have a limit as \( x \) approaches 1 from either side because the square root in the denominator implies that the values under the root must be positive to stay in the realm of real numbers. Since this condition can't be met, the function doesn't approach a single real number, illustrating why the limit does not exist.

Approaching Limits

Approaching limits is a foundational concept in calculus that involves determining the value that a function \( f(x) \) gets close to as the input \( x \) gets close to some number \( c \). The idea is not about where the function is at \( x=c \), but rather what value it is approaching. You can approach from the left (negative direction), the right (positive direction), or both sides for a two-sided limit.

For two-sided limits to exist, the one-sided limits must be the same from both directions. However, in our exercise, the function \( g(x) = \frac{x-3}{\sqrt{x^2 - 5x + 4}} \) does not have a two-sided limit as \( x \) approaches 1 because we've established the limit does not exist when approaching from either side. This discontinuity around \( x=1 \) is precisely why the limit, when approaching from both sides, cannot exist, highlighting the importance of understanding approaching limits to unravel the behavior of functions near points of interest.