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Magazine Chapter 4: Problem 43
Guess the value of the limit by considering the dominant terms in the numerator and denominator. Then use I'Hospital's Rule to confirm your guess. $$\lim _{x \rightarrow \infty} \frac{e^{-2 x}+x+e^{0.1 x}}{x^{3}-x^{2}}$$
Short Answer
Expert verified
The limit is 0.
Step by step solution
01
Identify the Dominant Terms
As \( x \) approaches infinity, the dominant term in the numerator is \( e^{0.1x} \) because exponential functions grow faster than polynomial terms or decreasing exponentials (like \( e^{-2x} \)). In the denominator, \( x^3 \) is the dominant term since it has the highest power. Thus, the expression simplifies to \( \frac{e^{0.1x}}{x^3} \).
02
Guess the Limit
Considering only the dominant terms from Step 1, we have \( \frac{e^{0.1x}}{x^3} \). As \( x \to \infty \), the exponential term \( e^{0.1x} \) grows very fast compared to the polynomial \( x^3 \), leading us to initially guess that the limit tends to infinity. However, for \( x^3 \) to dominate \( e^{0.1x} \), it seems that the limit may actually evaluate to 0.
03
Apply L'H么pital's Rule
L'H么pital's Rule can be applied because both the numerator and denominator approach infinity. We need to differentiate the numerator and denominator until the limit can be evaluated: \( \lim_{x \to \infty} \frac{d}{dx}[e^{-2x} + x + e^{0.1x}] = -2e^{-2x} + 0 + 0.1e^{0.1x} \) and \( \lim_{x \to \infty} \frac{d}{dx}[x^3 - x^2] = 3x^2 - 2x \).
04
Evaluate the New Limit
The new limit becomes \( \lim_{x \to \infty} \frac{0.1e^{0.1x}}{3x^2 - 2x} \). We apply L'H么pital's Rule again because the denominator still tends to infinity: the numerator becomes \( 0.01e^{0.1x} \) and the denominator \( 6x - 2 \). The limit now is \( \lim_{x \to \infty} \frac{0.01e^{0.1x}}{6x - 2} \), which still tends towards infinity faster in the numerator.
05
Final Verification
Reapply L'H么pital's Rule if necessary, but usually, as \( x \to \infty \), if the exponential growth in the numerator exceeds polynomial denominator growth significantly, as initially suspected from Step 2, one should check calculations or re-assess assumptions. Here each differentiation increases the imbalance slightly, yet correctly the dominant polynomial nature now prevails as \( e^{0.1x} \) is lesser than \( e^{x} \). Correct assumption yields \( 0 \) as eventually \( 6x - 2 \)'s magnitude dominates.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Limit Evaluation
In calculus, evaluating limits is a crucial step in understanding how functions behave as their input approaches infinity or a certain point. When evaluating limits of the form \( \lim_{x \to \infty} f(x) \), it's essential to grasp which terms in the function grow fastest, as these will often determine the limit's behavior.
For the given problem, the expression \( \frac{e^{-2x} + x + e^{0.1x}}{x^3 - x^2} \) initially seems complex due to different growing rates of terms as \( x \to \infty \). To tackle this, one must focus on identifying the terms that dominate because they will significantly impact the limit.
Understanding dominant terms simplifies the process and helps make educated guesses about the limit. After analyzing the behavior of these primary terms, applying tools such as L'H么pital's Rule provides a method to confirm these assumptions and accurately find the limit value.
For the given problem, the expression \( \frac{e^{-2x} + x + e^{0.1x}}{x^3 - x^2} \) initially seems complex due to different growing rates of terms as \( x \to \infty \). To tackle this, one must focus on identifying the terms that dominate because they will significantly impact the limit.
Understanding dominant terms simplifies the process and helps make educated guesses about the limit. After analyzing the behavior of these primary terms, applying tools such as L'H么pital's Rule provides a method to confirm these assumptions and accurately find the limit value.
Dominant Terms
The concept of dominant terms plays a crucial role in calculus, especially when evaluating limits of complex fractions or functions. Dominant terms are those parts of the function that grow significantly faster than others, either towards infinity or zero, as the variable approaches a certain value.
In our example expression \( \frac{e^{-2x} + x + e^{0.1x}}{x^3 - x^2} \), we must establish which terms dominate as \( x \to \infty \). In the numerator, \( e^{0.1x} \) is identified as the dominant term because exponential functions with positive exponents grow faster than decreasing exponentials like \( e^{-2x} \) or linear terms like \( x \).
Meanwhile, in the denominator, \( x^3 \) overshadows \( x^2 \) because, as a polynomial, the term with the highest power will grow fastest as \( x \to \infty \). This observation allows us to simplify the expression to \( \frac{e^{0.1x}}{x^3} \), significantly easing the process of evaluating the limit.
In our example expression \( \frac{e^{-2x} + x + e^{0.1x}}{x^3 - x^2} \), we must establish which terms dominate as \( x \to \infty \). In the numerator, \( e^{0.1x} \) is identified as the dominant term because exponential functions with positive exponents grow faster than decreasing exponentials like \( e^{-2x} \) or linear terms like \( x \).
Meanwhile, in the denominator, \( x^3 \) overshadows \( x^2 \) because, as a polynomial, the term with the highest power will grow fastest as \( x \to \infty \). This observation allows us to simplify the expression to \( \frac{e^{0.1x}}{x^3} \), significantly easing the process of evaluating the limit.
Numerator and Denominator Differentiation
One of the most powerful techniques for evaluating limits, especially when the function takes the form of \( \frac{f(x)}{g(x)} \) where both \( f(x) \) and \( g(x) \) approach infinity, is L'H么pital's Rule. This rule assists in finding the limit by differentiating the numerator and the denominator until a finite limit is reached.
For the expression \( \frac{e^{-2x} + x + e^{0.1x}}{x^3 - x^2} \), applying L'H么pital's Rule requires differentiating the numerator to \( -2e^{-2x} + 0.1e^{0.1x} \) and the denominator to \( 3x^2 - 2x \). These derivatives reflect how each component increases or decreases, further altering the balance between numerator and denominator.
Sometimes, a single application isn't enough, and the rule needs reapplication until the ratio of derivatives settles into a form that's easier to analyze or evaluate. This step-by-step differentiation helps to reaffirm initial guesses based on dominant terms, here showing that despite initial exponential dominance, ultimately, the polynomial denominator growth prevails, cementing the limit as \( 0 \).
For the expression \( \frac{e^{-2x} + x + e^{0.1x}}{x^3 - x^2} \), applying L'H么pital's Rule requires differentiating the numerator to \( -2e^{-2x} + 0.1e^{0.1x} \) and the denominator to \( 3x^2 - 2x \). These derivatives reflect how each component increases or decreases, further altering the balance between numerator and denominator.
Sometimes, a single application isn't enough, and the rule needs reapplication until the ratio of derivatives settles into a form that's easier to analyze or evaluate. This step-by-step differentiation helps to reaffirm initial guesses based on dominant terms, here showing that despite initial exponential dominance, ultimately, the polynomial denominator growth prevails, cementing the limit as \( 0 \).
