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Magazine Chapter 15: Problem 24
Evaluate the limit, if it exists. $$ \lim _{x \rightarrow 0} \frac{\sinh x-\sin x}{\sin ^{3} x} $$
Short Answer
Expert verified
The limit is 0.
Step by step solution
01
Analyze the Inner Exponents
Consider the expression inside the limit: \( x^{x^x} \). Let's denote \( y = x^x \), so we need to analyze the behavior of \( y \) as \( x \to 0^{+} \).
02
Simplify the Exponentiation
Rewrite \( y eq x \to 0^{+}(x^x) \) as \( x^{x^x} = x^y \). Now, consider \( x^x \). We know that \( x^x = e^{x \ln{x}} \).
03
Evaluate \( x \ln{x} \)
Analyze the exponent in \( e^{x \ln{x}} \) as \( x \rightarrow 0^{+} \). Since \( \ln{x} \longrightarrow -\infty \) and \( x \to 0 \), the product \( x \ln{x} \longrightarrow 0 \).
04
Conclude Behavior of \( x^x \)
Given \( e^{x \ln{x}} \rtarrow e^0 = 1 \), we have \( x^x \rightarrow 1 \) as \( x \rightarrow 0^{+} \).
05
Substitute Back to Original Expression
Recalling that \( x^{x^x} \) became \( x^y \) and \( x^x \rightarrow 1 \), we see that \( x^{1} = x \).
06
Evaluate the Limit Expression
Finally, as \( x \rightarrow 0^{+} \), the expression \( x \rightarrow 0 \). Hence, \( \lim_{x \rightarrow 0^{+}} x^{x^x} = \lim_{x \rightarrow 0^{+}} x = 0\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit Evaluation
Limits help us understand the behavior of a function as its input approaches a particular value. In this exercise, we're evaluating the limit of the function \( x^{x^{x}} \) as \( x \) approaches 0 from the positive side, symbolized as \( \rightarrow 0^{+} \). When approaching a complex limit, break down the expression into simpler components. Here, we simplify \( x^{x^{x}} \) step by step:
Understanding limits is essential as it lays a foundational step for further calculus concepts like derivatives and integrals.
- First, we analyze the innermost exponent \( y = x^x \).
- Then, we rewrite it using properties of exponents and logarithms.
- Next, we simplify \( x \to 0^{+} \) to observe how the components behave.
Understanding limits is essential as it lays a foundational step for further calculus concepts like derivatives and integrals.
Exponential Functions
Exponential functions are pivotal in calculus. They have the form \( f(x) = a^x \), where \( a \) is a positive constant. In this problem, \( x^x \) shows the behavior of a function where both base and exponent are variables.
When simplifying \( x^{x^x} \), we noted:
Consider these key characteristics of exponential functions:
When simplifying \( x^{x^x} \), we noted:
- \( x^x = e^{x \text{ln} x} \)
Consider these key characteristics of exponential functions:
- They grow rapidly as \( x \) increases.
- When \( x \) is small or negative, they approach 0 but never reach it.
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions, typically written as \( f(x) = \text{ln}(x) \), where \( \text{ln} \) represents the natural logarithm. In our problem, logarithms helped simplify the exponent in \( x^{x} \):
Remember these aspects of logarithmic functions:
- By rewriting \( x^{x} \) as \( e^{x \text{ln} x} \), we could leverage properties of both exponential and logarithmic functions.
- We observed that \( \text{ln}(x) \) approaches \( -\text{infty} \) as \( x \to 0^{+} \).
Remember these aspects of logarithmic functions:
- They grow slowly compared to polynomials and exponential functions.
- They are undefined for non-positive values of \( x \).
