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Magazine Chapter 1: Problem 62
Prove that if \(\lim _{x \rightarrow a} g(x)=0\) and \(\lim _{x \rightarrow a} f(x)\) exists and is not \(0,\) then $$\lim _{x \rightarrow a} \frac{f(x)}{g(x)} \quad$$ does not exist
Short Answer
Expert verified
The limit does not exist because \( \frac{f(x)}{g(x)} \) becomes unbounded as \( x \to a \).
Step by step solution
01
Understand the Given Conditions
We are given two limits: \( \lim_{x \to a} g(x) = 0 \) and \( \lim_{x \to a} f(x) \) exists and is non-zero. Our goal is to show that \( \lim_{x \to a}\frac{f(x)}{g(x)} \) does not exist.
02
Apply Limit Definition
According to the definition of a limit, \( \lim_{x \to a} g(x) = 0 \) means that as \( x \) approaches \( a \), \( g(x) \) gets arbitrarily close to 0. For \( \lim_{x \to a} f(x) = L eq 0 \), \( f(x) \) approaches a non-zero value \( L \) as \( x \) approaches \( a \).
03
Examine the Expression \(\frac{f(x)}{g(x)}\)
Consider \(\frac{f(x)}{g(x)}\) as \(x\) approaches \(a\). Since \(g(x)\) approaches 0, the expression \(\frac{1}{g(x)}\) will grow without bound, leading to the possibility of division by zero or an undefined expression.
04
Behavior of \(\frac{f(x)}{g(x)}\) as \(x \rightarrow a\)
Because \( f(x) \) approaches a non-zero value and \( g(x) \) approaches zero, the expression \( \frac{f(x)}{g(x)} \) will tend to infinity or negative infinity. This unbounded behavior indicates that the limit does not stabilize to any finite value.
05
Conclusion on the Limit
Since \(\frac{f(x)}{g(x)}\) becomes unbounded as \( x \to a \), and it does not approach any finite limit, we conclude that \( \lim_{x \to a}\frac{f(x)}{g(x)} \) does not exist.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit definition
The concept of a limit in calculus is fundamental when it comes to analyzing functions as they approach specific points. When we write \( \lim_{x \to a} f(x) = L \), it means that as \( x \) gets closer and closer to \( a \), the value of \( f(x) \) approaches \( L \).
This concept is crucial in understanding how functions behave near certain points, even if they are not defined exactly at those points.
In this context, approaching a point does not imply that the function's value will reach \( L \) at \( x = a \); rather, it's about the behavior of \( f(x) \) as we get infinitely close to \( a \).
This concept is crucial in understanding how functions behave near certain points, even if they are not defined exactly at those points.
In this context, approaching a point does not imply that the function's value will reach \( L \) at \( x = a \); rather, it's about the behavior of \( f(x) \) as we get infinitely close to \( a \).
- If the limit exists, \( f(x) \) should settle into a consistent value near \( a \).
- However, in some cases, instead of stabilizing, a function can exhibit erratic behavior as it nears a point, leading to a conclusion that the limit does not exist.
Division by zero
One of the dilemmas in calculus is managing instances where division by zero occurs. When we have a scenario where \( g(x) \to 0 \) as \( x \to a \) and \( f(x) \) remains non-zero, \( \frac{f(x)}{g(x)} \) becomes problematic.
Division by zero is a mathematical no-go since it leads to undefined results. Attempting to divide something by a quantity approaching zero results in the expression growing without bounds.
This is crucial to our understanding because:
Division by zero is a mathematical no-go since it leads to undefined results. Attempting to divide something by a quantity approaching zero results in the expression growing without bounds.
This is crucial to our understanding because:
- As the denominator gets closer to zero, the fraction's value can become exceedingly large or small, showing undefined or infinite behavior.
- This unbounded behavior underscores the necessity of being cautious around points where the denominator tends to zero.
Unbounded behavior
Unbounded behavior in mathematical functions refers to situations where a function does not approach any limit, but instead grows excessively large. When the denominator of a fraction like \( \frac{f(x)}{g(x)} \) becomes infinitely small (as \( g(x) \to 0 \)), the fraction itself can grow extremely large.
In calculus, unbounded behavior indicates that the function is moving towards infinity or negative infinity, rather than settling into a finite value.
This is particularly crucial when noting that:
When we observe unbounded behavior, like the one from \( \frac{f(x)}{g(x)} \) as \( x \to a \), it's generally concluded that the limit does not exist.
In calculus, unbounded behavior indicates that the function is moving towards infinity or negative infinity, rather than settling into a finite value.
This is particularly crucial when noting that:
- Such behavior means that the limit is non-existent, as no specific value is approached.
- The function doesn't converge to a single number, which is a red flag for the limit鈥檚 existence.
When we observe unbounded behavior, like the one from \( \frac{f(x)}{g(x)} \) as \( x \to a \), it's generally concluded that the limit does not exist.
Calculus proof
In calculus, a proof is a logical explanation demonstrating the truth of a statement or theorem. To prove that \( \lim_{x \to a} \frac{f(x)}{g(x)} \) does not exist, we use specific limits' properties and behaviors.
The process generally involves:
The essence of a calculus proof in this context revolves around demonstrating that because the denominator approaches zero and the numerator remains non-zero, the expression becomes unbounded.
This logically leads to concluding that the original limit, \( \lim_{x \to a} \frac{f(x)}{g(x)} \), cannot stabilize to any finite or predictable number, hence does not exist.
The process generally involves:
- Analyzing each component of the expression, like \( f(x) \) and \( g(x) \), and understanding their behavior around the point \( a \).
- Using this understanding to explore the behavior of their ratio \( \frac{f(x)}{g(x)} \).
The essence of a calculus proof in this context revolves around demonstrating that because the denominator approaches zero and the numerator remains non-zero, the expression becomes unbounded.
This logically leads to concluding that the original limit, \( \lim_{x \to a} \frac{f(x)}{g(x)} \), cannot stabilize to any finite or predictable number, hence does not exist.
