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Chapter 2: Problem 45

We say that \(\lim _{x \rightarrow a^{+}} f(x)=-\infty\) if for any negative number \(N,\) there exists \(\delta>0\) such that $$f(x)

Short Answer

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Question: Write the formal definitions for the following limit types for a function using the given definition for the right-hand limit when it approaches negative infinity: a. Right-hand limit approaching positive infinity. b. Overall limit approaching negative infinity. c. Left-hand limit approaching positive infinity. Answer: a. The formal definition of the right-hand limit approaching positive infinity is: \(\lim_{x \rightarrow a^{+}} f(x) = \infty\) if for any positive number \(P\), there exists \(\delta > 0\) such that \(f(x) > P\) whenever \(a < x < a+\delta\). b. The formal definition of the overall limit approaching negative infinity is: \(\lim_{x \rightarrow a} f(x) = -\infty\) if for any negative number \(N\), there exists \(\delta > 0\) such that \(f(x) < N\) whenever \(0 < |x-a| < \delta\). c. The formal definition of the left-hand limit approaching positive infinity is: \(\lim_{x \rightarrow a^{-}} f(x) = \infty\) if for any positive number \(P\), there exists \(\delta > 0\) such that \(f(x) > P\) whenever \(a-\delta < x < a\).

Step by step solution

01

Formal Definition of \(\lim_{x \rightarrow a^{+}} f(x) = \infty\)

The limit \(\lim_{x \rightarrow a^{+}} f(x) = \infty\) if for any positive number \(P\), there exists \(\delta > 0\) such that $$f(x) > P \quad \text{whenever} \quad a < x < a+\delta$$ b. Overall limit approaching negative infinity:
02

Formal Definition of \(\lim_{x \rightarrow a} f(x) = -\infty\)

The limit \(\lim_{x \rightarrow a} f(x) = -\infty\) if for any negative number \(N\), there exists \(\delta > 0\) such that $$f(x) < N \quad \text{whenever} \quad 0 < |x-a| < \delta$$ c. Left-hand limit approaching positive infinity:
03

Formal Definition of \(\lim_{x \rightarrow a^{-}} f(x) = \infty\)

The limit \(\lim_{x \rightarrow a^{-}} f(x) = \infty\) if for any positive number \(P\), there exists \(\delta > 0\) such that $$f(x) > P \quad \text{whenever} \quad a-\delta < x < a$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Formal Definition of Limits
In calculus, the formal definition of a limit helps us understand the behavior of a function as its input approaches a certain value. Understanding limits is crucial in calculus as it forms the foundation for the concept of derivation and integration. When talking about \(\lim_{x \rightarrow a} f(x) = L\), it means that as \(x\) gets closer to \(a\), the function \(f(x)\) approaches the value \(L\). This idea can be extended for when \(f(x)\) tends towards infinity.

When considering \(\lim_{x \rightarrow a^{+}} f(x) = \infty\), it means that for any chosen large positive number \(P\), a point can be found where, for values of \(x\) just bigger than \(a\), \(f(x)\) becomes larger than \(P\). Similarly, for a negative infinity limit, when \(\lim_{x \rightarrow a} f(x) = -\infty\), for every negative number \(N\), there's a small distance such that as \(x\) nears \(a\), \(f(x)\) dips below \(N\). These formal statements help in making the abstract concept of limits concrete through \(\delta\) and \(\varepsilon\) conditions.
  • Establishing how close \(x\) should be to \(a\) establishes rigor in how we "approach" a limit.
  • It helps maintain the precision required when dealing with infinite behaviors of functions.
  • This definition can serve as a tool for reasoning when limits aren't straightforward to simplify.
Limits Approaching Infinity
Exploring limits as \(x\) moves towards infinity has some special implications. When \(x\) approaches a point like infinity, \(f(x)\) can either head to a particular number or deviate towards infinity itself. Mathematically, \(\lim_{x \rightarrow \infty} f(x) = L\) means that as \(x\) grows without bound, \(f(x)\) approaches a fixed number, \(L\).

For instances where the function itself grows infinitely, you might see a case where \(\lim_{x \rightarrow a} f(x) = \infty\). Here, "\(a\)" could be infinity itself. This occurs in functions like exponential growth, where they skyrocket dramatically as the variable grows large. Especially for rational functions, this could lead to vertical asymptotes鈥攑laces where the graph aggressively climbs up or plummets.
  • Understanding behavior as the inputs extend to infinity is crucial for grasping unbounded growth or decrease.
  • Real-world phenomena like population growth or radioactive decay use concepts of approaching infinity.
  • Limits and asymptotes provide a snapshot of what's happening to a graph far out of the normal viewing window.
One-Sided Limits
One-sided limits provide a snapshot of how a function behaves as it approaches a particular value \(a\) just from one side. They bring extra detail by specifying if \(x\) is approaching \(a\) from the left (notated as \(a^{-}\)) or from the right (denoted as \(a^{+}\)).

Specifically, a right-hand limit \(\lim_{x \rightarrow a^{+}} f(x) = L\) looks at what happens as \(x\) approaches \(a\) from numbers bigger than \(a\). Whereas its counterpart, left-hand limit, \(\lim_{x \rightarrow a^{-}} f(x) = L\), observes the behavior with \(x\) approaching \(a\) from numbers smaller than \(a\).
  • These concepts are vital when functions have discontinuities or jump points.
  • One-sided limits are used heavily when functions behave differently before and after a certain point.
  • They help to determine the existence of an overall limit by comparing values approached from either side.

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Most popular questions from this chapter

Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=2^{x}$$

Determine the end behavior of the following transcendental functions by evaluating appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=|\ln x|$$

A monk set out from a monastery in the valley at dawn. He walked all day up a winding path, stopping for lunch and taking a nap along the way. At dusk, he arrived at a temple on the mountaintop. The next day, the monk made the return walk to the valley, leaving the temple at dawn, walking the same path for the entire day, and arriving at the monastery in the evening. Must there be one point along the path that the monk occupied at the same time of day on both the ascent and descent? (Hint: The question can be answered without the Intermediate Value Theorem.) (Source: Arthur Koestler, The Act of Creation.)

Analyzing infinite limits graphically Graph the function \(y=\sec x \tan x\) with the window \([-\pi, \pi] \times[-10,10] .\) Use the graph to analyze the following limits. a. \(\lim _{x \rightarrow \pi / 2^{+}} \sec x \tan x\) b. \(\lim _{x \rightarrow \pi / 2^{-}} \sec x \tan x\) c. \(\lim _{x \rightarrow-\pi / 2^{+}} \sec x \tan x\) d. \(\lim _{x \rightarrow-\pi / 2^{-}} \sec x \tan x\)

Prove Theorem 11: If \(g\) is continuous at \(a\) and \(f\) is continuous at \(g(a),\) then the composition \(f \circ g\) is continuous at \(a .\) (Hint: Write the definition of continuity for \(f\) and \(g\) separately; then, combine them to form the definition of continuity for \(\left.f^{\circ} g .\right)\)
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