Math Problem Statement

Solution

The given expression is:

limx(1x+1ex)\lim_{{x \to \infty}} \left( 1 - \frac{x+1}{e^x} \right)

To solve this limit, let's analyze the term inside the limit.

  1. Consider the term x+1ex\frac{x+1}{e^x}.
  2. As xx \to \infty, exe^x grows exponentially faster than x+1x+1.

Thus, x+1ex0\frac{x+1}{e^x} \to 0 as xx \to \infty.

Now, substituting this back into the original expression:

limx(1x+1ex)=10=1\lim_{{x \to \infty}} \left( 1 - \frac{x+1}{e^x} \right) = 1 - 0 = 1

Therefore, the limit is:

1\boxed{1}

Would you like any details or have any questions?

Here are 8 relative questions that expand on the information:

  1. What happens to xex\frac{x}{e^x} as xx \to \infty?
  2. How does the behavior of exe^x compare to polynomial functions as xx \to \infty?
  3. What are some other common limits involving exe^x as xx \to \infty?
  4. How do you use L'Hôpital's Rule to evaluate limits of indeterminate forms?
  5. Can you explain why exponential functions grow faster than polynomial functions?
  6. How would the limit change if the numerator were a different polynomial, e.g., x2+xex\frac{x^2 + x}{e^x}?
  7. What is the significance of the constant ee in calculus and analysis?
  8. Can this limit be evaluated using series expansion?

Tip: When evaluating limits at infinity, compare the rates of growth of the functions involved to determine which terms dominate.

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Math Problem Analysis

Mathematical Concepts

Limits
Exponential Functions
Indeterminate Forms

Formulas

L'Hôpital's Rule

Theorems

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Suitable Grade Level

Advanced High School / Undergraduate Level