Problem 53 - Calculus Early Transcendentals

Question

Let \(f(x)=\frac{x^{2}-7 x+12}{x-a}\) a. For what values of \(a,\) if any, does \(\lim _{x \rightarrow a^{+}} f(x)\) equal a finite number? b. For what values of \(a,\) if any, does \(\lim _{x \rightarrow a^{+}} f(x)=\infty ?\) c. For what values of \(a\), if any, does \(\lim _{x \rightarrow a^{+}} f(x)=-\infty ?\)

Step-by-Step Solution

Answer

Question: For the function \(f(x) = \frac{x^2 - 7x + 12}{x-a}\), determine the values of \(a\) for which the limit as \(x\) approaches \(a\) from the right side is a) finite, b) infinity, and c) negative infinity. Answer: a) The limit is finite for all values of \(a\). b) The limit is infinity for \(a > 4\). c) The limit is negative infinity for \(3 < a < 4\).

Step 1: Factorize the numerator

We need to factorize the quadratic \(x^2 - 7x + 12\). To do this, we find two numbers whose product equals 12 and whose sum equals -7. The numbers are -3 and -4. So, the factorization of the numerator is \((x-3)(x-4)\), and the function \(f(x)\) becomes: \(f(x) = \frac{(x-3)(x-4)}{x-a}\)

Step 2: Find the limit for each case

For each case, we analyze the limit of the function \(f(x)\) as \(x\) approaches \(a^+\). a) We need to find the values of \(a\) for which the limit is finite. The limit is finite if the function is continuous at \(x=a\). Since the numerator is always a finite value, it is important that the denominator is not equal to zero. Therefore, we must have \(x \neq a\). The function is continuous for all \(x\) except \(a\). Thus, the limit is finite for all values of \(a\). b) We need to find the values of \(a\) for which the limit equals infinity. The limit will equal infinity if the denominator tends to zero and the numerator is positive. Since the denominator is \(x-a\), it tends to zero as \(x \rightarrow a^+\). As for the numerator, we analyze when \((x-3)(x-4)\) is positive. Case 1: \(x>4 \Rightarrow (x-3)(x-4) > 0\) Case 2: \(3 0\) Since we are considering the limit as \(x \rightarrow a^+\), we focus on Case 1 where \(a>4\): \(\lim_{x \rightarrow a^+} f(x) = \frac{(x-3)(x-4)}{x-a} = \infty\) c) We need to find the values of \(a\) for which the limit equals negative infinity. The limit will equal negative infinity if the denominator tends to zero and the numerator is negative. Since the denominator is \(x-a\), it tends to zero as \(x \rightarrow a^+\). Referring back to the analyzed cases for the numerator, when \(3

Step 3: Summarize the results

Based on the analysis, we have the following results: a) The limit equals a finite number for all values of \(a\). b) The limit equals infinity for \(a > 4\). c) The limit equals negative infinity for \(3 < a < 4\).

Key Concepts

Limit of a Function

The concept of a limit of a function is foundational in calculus. It's about understanding what happens to the value of a function as the input value approaches a certain point, not necessarily reaching that point.

When considering the limit of a function as the input, denoted as 'x', approaches a particular value 'a', we are essentially peeking at the function's behavior around 'a'. If you can predict the function's value as 'x' gets infinitely close to 'a', you've found the limit.

For instance, in the provided exercise, we look at the function \( f(x) = \frac{x^2 - 7x + 12}{x-a} \). To determine the limit as \( x \) approaches 'a' from the right (denoted as \( x \rightarrow a^+ \)), we focus on the behavior of the function when 'x' is slightly greater than 'a'. If \( x \) is not equal to 'a', it means the function will not go to infinity due to division by zero. Thus, for all values of 'a' different from the roots of the numerator, the function will approach a finite limit. This intricate dance between 'x' and 'a' produces fascinating outcomes that are crucial to the subject of calculus.

Continuity

When we talk about continuity in calculus, we're talking about a smooth, unbroken function without any jumps, holes, or asymptotes. A function is continuous at a point 'x=a' if the limit as 'x' approaches 'a' is equal to the function's value at 'x=a': \( \text{lim}_{x \rightarrow a} f(x) = f(a) \).

For a function to be continuous over an interval, it has to be continuous at every point within that interval. Let's apply this to the exercise function \( f(x) \). We observed that since the numerator of the function is factorizable to \( (x-3)(x-4) \), the function is continuous at all points except where the denominator is zero, which would be 'x=a'.

Therefore, the function is not continuous at 'x=a', resulting in the limit being finite for all values of 'a' not equal to 3 or 4, where the numerator would be zero too. Understanding continuity is essential as it ensures the function can be graphed without lifting the pencil from the paper – a seamless curve representing the function.

Infinite Limits

Moving on to infinite limits, these occur when the value of the function increases or decreases without bound as 'x' approaches a specific value. It's a situation where a function doesn't settle on a finite limit but instead heads towards infinity (or negative infinity).

In the provided exercise, we're asked to determine when \( f(x) \) heads off to infinity or negative infinity as 'x' comes from the right side of 'a'. Infinite limits are intriguing because they signify a vertical asymptote in the graph – a line that the function can get infinitely close to but never actually touch or cross.

We find that the function shoots off to positive infinity when 'a' is greater than 4 and plunges into negative infinity for values of 'a' between 3 and 4. This occurs because the values of 'x' make the fraction inside \( f(x) \) have a very tiny denominator and a positive (or negative) numerator.

Infinite limits reflect significant traits of function behavior and are pivotal when analyzing graphs for vertical asymptotes. They're also a dramatic illustration of calculus's power to describe not just gentle slopes, but also the most sudden of mathematical cliffs.

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Problem 53

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Key Concepts

Limit of a Function

The concept of a limit of a function is foundational in calculus. It's about understanding what happens to the value of a function as the input value approaches a certain point, not necessarily reaching that point.

When considering the limit of a function as the input, denoted as 'x', approaches a particular value 'a', we are essentially peeking at the function's behavior around 'a'. If you can predict the function's value as 'x' gets infinitely close to 'a', you've found the limit.

For instance, in the provided exercise, we look at the function \( f(x) = \frac{x^2 - 7x + 12}{x-a} \). To determine the limit as \( x \) approaches 'a' from the right (denoted as \( x \rightarrow a^+ \)), we focus on the behavior of the function when 'x' is slightly greater than 'a'. If \( x \) is not equal to 'a', it means the function will not go to infinity due to division by zero. Thus, for all values of 'a' different from the roots of the numerator, the function will approach a finite limit. This intricate dance between 'x' and 'a' produces fascinating outcomes that are crucial to the subject of calculus.

Continuity

When we talk about continuity in calculus, we're talking about a smooth, unbroken function without any jumps, holes, or asymptotes. A function is continuous at a point 'x=a' if the limit as 'x' approaches 'a' is equal to the function's value at 'x=a': \( \text{lim}_{x \rightarrow a} f(x) = f(a) \).

For a function to be continuous over an interval, it has to be continuous at every point within that interval. Let's apply this to the exercise function \( f(x) \). We observed that since the numerator of the function is factorizable to \( (x-3)(x-4) \), the function is continuous at all points except where the denominator is zero, which would be 'x=a'.

Therefore, the function is not continuous at 'x=a', resulting in the limit being finite for all values of 'a' not equal to 3 or 4, where the numerator would be zero too. Understanding continuity is essential as it ensures the function can be graphed without lifting the pencil from the paper – a seamless curve representing the function.

Infinite Limits

Moving on to infinite limits, these occur when the value of the function increases or decreases without bound as 'x' approaches a specific value. It's a situation where a function doesn't settle on a finite limit but instead heads towards infinity (or negative infinity).

In the provided exercise, we're asked to determine when \( f(x) \) heads off to infinity or negative infinity as 'x' comes from the right side of 'a'. Infinite limits are intriguing because they signify a vertical asymptote in the graph – a line that the function can get infinitely close to but never actually touch or cross.

We find that the function shoots off to positive infinity when 'a' is greater than 4 and plunges into negative infinity for values of 'a' between 3 and 4. This occurs because the values of 'x' make the fraction inside \( f(x) \) have a very tiny denominator and a positive (or negative) numerator.

Infinite limits reflect significant traits of function behavior and are pivotal when analyzing graphs for vertical asymptotes. They're also a dramatic illustration of calculus's power to describe not just gentle slopes, but also the most sudden of mathematical cliffs.