Problem 101 - Calculus

Question

Which of the following statements is a direct consequence of the statement: "If \(f\) and \(g\) are continuous at \(x=a\) and \(g(a) \neq 0\) then \(f / g\) is continuous at \(x=a ?\) (a) If \(f\) and \(g\) are continuous at \(x=a\) and \(f(a) \neq 0\) then \(g / f\) is continuous at \(x=a\) (b) If \(f\) and \(g\) are continuous at \(x=a\) and \(g(a)=0\) then \(f / g\) is not continuous at \(x=a\) (c) If \(f, g,\) are continuous at \(x=a,\) but \(f / g\) is not continuous at \(x=a,\) then \(g(a)=0\) (d) If \(f\) and \(f / g\) are continuous at \(x=a\) and \(g(a) \neq 0\) then \(g\) is continuous at \(x=a\)

Step-by-Step Solution

Answer

Option (c) is correct; \(g(a) = 0\) implies \(\frac{f}{g}\) is not continuous.

Step 1: Understanding the Given Statement

The given statement says: "If \(f\) and \(g\) are continuous at \(x=a\) and \(g(a) eq 0\), then \(\frac{f}{g}\) is continuous at \(x=a\)." This implies that for \(\frac{f}{g}\) to be continuous, both \(f\) and \(g\) must be continuous at \(x=a\), with \(g(a)\) also being non-zero, ensuring the denominator does not become undefined.

Step 2: Analyzing Each Option

We analyze each provided option:- Option (a) states that if \(f\) and \(g\) are continuous at \(x=a\) and \(f(a) eq 0\), then \(\frac{g}{f}\) is continuous at \(x=a\). This statement is incorrect as it swaps the conditions on \(f\) and \(g\).- Option (b) says if \(f\) and \(g\) are continuous at \(x=a\) and \(g(a)=0\), then \(\frac{f}{g}\) is not continuous at \(x=a\). This may be true, but it is not a direct consequence of the given statement.- Option (c) suggests that if \(f\) and \(g\) are continuous at \(x=a\), but \(\frac{f}{g}\) is not continuous at \(x=a\), then \(g(a)=0\). This is a direct logical consequence as having \(g(a) ≠ 0\) is essential for continuity of \(\frac{f}{g}\).- Option (d) claims that if \(f\) and \(\frac{f}{g}\) are continuous at \(x=a\), with \(g(a) ≠ 0\), then \(g\) is continuous at \(x=a\). But continuity of \(g\) is already assumed in the statement.

Step 3: Conclusion

The statement that best aligns as a direct consequence of the given statement is option (c). The essence of the given statement is that \(g(a) eq 0\) is essential for ensuring the continuity of \(\frac{f}{g}\) along with \(f\) and \(g\) being continuous, making \(c)\) correct as it is directly derived from the negation of the hypothesis.

Key Concepts

Continuous functions

In calculus, continuous functions are the cornerstone of much of what is done in the study of limits and derivatives. A function is said to be continuous at a point if there are no abrupt jumps, breaks, or holes in the graph of the function at that point. This intuitively means that you can draw the graph of the function at that specific point without lifting your pencil off the paper.
To be mathematically precise, a function \( f(x) \) is continuous at a point \( x = a \) if the following three conditions are met:

• \( f(a) \) is defined.
• The limit of \( f(x) \) as \( x \) approaches \( a \) exists.
• The limit of \( f(x) \) as \( x \) approaches \( a \) is equal to \( f(a) \).

These conditions ensure that the function behaves well around the point of interest, and its value at the point precisely equals the limit of the function at that point.
Understanding continuous functions is essential as it aids in solving many problems involving limits and operations on functions.

Limits

The concept of limits is fundamental in calculus and underlies the definition of derivatives and integrals. Simply put, a limit is the value that a function \( f(x) \) approaches as the input \( x \) approaches a certain value \( a \).
Let's use the notation to be more specific. The expression \( \lim_{{x \to a}} f(x) = L \) means "the limit of \( f(x) \) as \( x \) approaches \( a \) is equal to \( L \)."
There are several ways limits can be approached:

• From the left: Limit as \( x \) approaches \( a \) from values less than \( a \), denoted as \( \lim_{{x \to a^-}} f(x) \).
• From the right: Limit as \( x \) approaches \( a \) from values greater than \( a \), denoted as \( \lim_{{x \to a^+}} f(x) \).
• Two-sided limit: Only exists if both one-sided limits are equal.

Limits are integral not just for understanding the behavior of functions near a point, but also for the formulation of derivatives and for evaluating integrals. They help in defining the pointwise convergence of functions and understanding how functions can be manipulated through operations like addition, multiplication, and finding quotients.

Rational functions

Rational functions play an influential role in calculus problems due to their inherent properties and their ability to model a plethora of real-world scenarios. A rational function is any function that can be expressed as the ratio of two polynomials, expressed in general form as:
\[R(x) = \frac{p(x)}{q(x)}\]where \( p(x) \) and \( q(x) \) are polynomials and \( q(x) eq 0 \).
These functions have distinct characteristics:

• **Domain:** Since the denominator \( q(x) \) cannot be zero, the domain of a rational function excludes those points that make the denominator zero.
• **Continuity:** They are continuous on their domain, meaning they do not have any breaks or jumps except possibly at the points where the denominator is zero.
• **Asymptotes:** Rational functions can have horizontal, vertical, or even oblique asymptotes depending on the highest degree of their numerator and denominator polynomials.

Understanding rational functions is crucial, particularly when assessing their limits, as this can directly impact the continuity of the function. These functions serve as prime examples when studying limits and continuity due to their behavior near points where \( q(x) = 0 \), which can result in undefined points or discontinuities in the graph.