Textbook Question
Evaluate the following limits in two different ways: with and without l’Hôpital’s Rule.
lim_x→∞ (2x⁵ - x + 1) / (5x⁶ + x)
4. Applications of Derivatives
Differentials
4:31 minutesProblem 4.R.79
Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_x→∞ (1 - (3/x))ˣ
Verified step by step guidance1
Recognize that the limit \( \lim_{x \to \infty} \left(1 - \frac{3}{x}\right)^x \) is of the indeterminate form \(1^\infty\). This suggests the use of the exponential limit property.
Rewrite the expression using the natural exponential function: \( \lim_{x \to \infty} \left(1 - \frac{3}{x}\right)^x = \lim_{x \to \infty} e^{x \ln\left(1 - \frac{3}{x}\right)} \).
Focus on evaluating the exponent \( \lim_{x \to \infty} x \ln\left(1 - \frac{3}{x}\right) \). This is still an indeterminate form \(0 \cdot (-\infty)\).
Use the approximation \( \ln(1 + u) \approx u \) for small \(u\), so \( \ln\left(1 - \frac{3}{x}\right) \approx -\frac{3}{x} \). Substitute this into the limit: \( \lim_{x \to \infty} x \left(-\frac{3}{x}\right) = \lim_{x \to \infty} -3 \).
Conclude that the original limit is \( e^{-3} \) by substituting back into the exponential form: \( \lim_{x \to \infty} e^{x \ln\left(1 - \frac{3}{x}\right)} = e^{-3} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior at points where it may not be explicitly defined, such as at infinity or discontinuities. Evaluating limits is crucial for determining the continuity and differentiability of functions.
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Exponential Functions
Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a positive constant. They exhibit rapid growth or decay and are characterized by their unique property that the rate of change is proportional to the function's value. Understanding their behavior, especially as x approaches infinity, is essential for evaluating limits involving exponential expressions.
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l'Hôpital's Rule
l'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits, especially in complex expressions.
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