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Chapter 6: Problem 11

Evaluate each limit (or state that it does not exist). $$ \lim _{a \rightarrow-\infty} e^{2 a} $$

Short Answer

Expert verified
The limit is 0.

Step by step solution

01

Understanding the Expression

The limit expression we need to evaluate is \( \lim _{a \rightarrow -\infty} e^{2a} \). This expression contains the exponential function \( e^{2a} \), where the base is the mathematical constant \( e \approx 2.718 \), and the expression inside is \( 2a \). As \( a \) approaches negative infinity, we need to evaluate how \( 2a \) affects the entire expression.
02

Analyzing \( 2a \) as \( a \to -\infty \)

When \( a \rightarrow -\infty \), the expression \( 2a \) also approaches \( -\infty \) because multiplying a negative number (\( a \)) by a positive constant (2) yields a more negative number. Therefore, as \( a \rightarrow -\infty \), \( 2a \rightarrow -\infty \).
03

Evaluating the Exponential Function

The behavior of the exponential function \( e^x \) as \( x \to -\infty \) is critical here. As \( x \to -\infty \), \( e^x \to 0 \). This is because the exponential function decreases rapidly towards zero as its exponent becomes a large negative number.
04

Calculating the Limit

Given the behavior of \( e^x \), when \( x = 2a \) and \( 2a \rightarrow -\infty \), the function \( e^{2a} \rightarrow 0 \). Hence, the limit is \( 0 \).

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Key Concepts

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

Exponential Functions
Exponential functions are a fundamental concept in mathematics. These functions have the form \( e^x \), where \( e \) is a constant approximately equal to 2.718, known as Euler's number. This constant is the base of the natural logarithm and plays a critical role in growth processes. Exponential functions are characterized by rapid growth or decay:
  • If the exponent is positive, the function grows rapidly as the variable increases.
  • If the exponent is negative, the function decreases rapidly towards zero.

For example, when you deal with an exponential function like \( e^{2a} \):
  • As \( a \) increases, \( 2a \) becomes more positive, and \( e^{2a} \) grows large.
  • Conversely, when \( a \) turns negative and gets larger in magnitude, \( 2a \) decreases, driving \( e^{2a} \) towards zero.
Thus, understanding exponential functions is essential when evaluating limits involving exponential terms.
Infinite Limits
Infinite limits occur when a variable approaches infinity or negative infinity, affecting the overall behavior of a function. As variables approach these extreme values, functions can either shoot towards an infinite value, zero, or sometimes become undefined.

In the context of exponential functions like \( e^{2a} \):
  • As \( a \to \infty \), the exponent \( 2a \) also tends to infinity, causing \( e^{2a} \) to grow infinitely large.
  • Conversely, as \( a \to -\infty \), \( 2a \) also approaches \( -\infty \), which results in the exponential function \( e^{2a} \) tending towards zero.

Infinite limits provide valuable insights into how functions behave as their inputs grow without restraint.
Evaluating Limits
Evaluating limits is a critical skill in calculus, allowing us to understand the behavior of functions at boundaries, especially when direct substitution is not possible. To evaluate a limit:

  • Consider the behavior of the function as the variable approaches a given value.
  • For exponential functions, recognize how rapid growth or decay impacts the function's approach to that limit.

In the exercise \( \lim_{a \to -\infty} e^{2a} \), observe how the function trends towards zero as \( a \) becomes a larger negative number. This analysis demonstrates a fundamental understanding of limits in the context of exponential decay. Evaluating such limits involves recognizing patterns and utilizing the inherent properties of exponential functions.

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Most popular questions from this chapter

Each equation follows from the integration by parts formula by replacing \(u\) by \(f(x)\) and \(v\) by a particular function. What is the function \(v\) ? $$ \int f(x) \frac{1}{x} d x=f(x) \ln x-\int \ln x f^{\prime}(x) d x $$

Evaluate each definite integral using integration by parts. (Leave answers in exact form.) $$ \int_{0}^{\ln 4} t e^{t} d t $$

Suppose that you now have $$\$ 2000$$, you expect to save an additional $$\$ 6000$$ during each year, and all of this is deposited in a bank paying \(4 \%\) interest compounded continuously. Let \(y(t)\) be your bank balance (in thousands of dollars) \(t\) years from now. a. Write a differential equation that expresses the fact that your balance will grow by 6 (thousand dollars) and also by \(4 \%\) of itself. [Hint: See Example 7.] b. Write an initial condition to say that at time zero the balance is 2 (thousand dollars). c. Solve your differential equation and initial condition. d. Use your solution to find your bank balance \(t=20\) years from now.

$$ \text { Use integration by parts to find each integral. } $$ $$ \int x^{5} \ln x d x $$

Let \(y(t)\) be the size of a colony of bacteria after \(t\) hours. a. Write a differential equation that says that the rate of growth of the colony is equal to eight times the three-fourths power of its present size. b. Write an initial condition that says that at time zero the colony is of size 10,000 . c. Solve the differential equation and initial condition. d. Use your solution to find the size of the colony at time \(t=6\) hours.
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