Lim x→0 (e^x - 1 - x)/x^2

Definition

The limit lim x→0 (e^x - 1 - x)/x^2 describes the behavior of the function as x approaches 0. This limit evaluates the rate of change of the function near zero, specifically how the expression behaves as it approaches a point where direct substitution leads to an indeterminate form. Understanding this limit is crucial for applying L'Hôpital's Rule, which helps simplify the evaluation of limits involving indeterminate forms.

5 Must Know Facts For Your Next Test

1. When substituting x = 0 directly into (e^x - 1 - x)/x^2, you get the indeterminate form 0/0, which requires further analysis to resolve.
2. Applying L'Hôpital's Rule, you differentiate the numerator and denominator repeatedly until you can evaluate the limit without encountering indeterminate forms.
3. After applying L'Hôpital's Rule twice, the limit evaluates to 1/2, which indicates the behavior of the function as x approaches 0.
4. This limit is related to the Taylor series expansion of e^x around x = 0, which provides insight into how e^x behaves near this point.
5. Understanding this limit helps with more complex calculus problems involving exponential functions and their approximations.

Review Questions

• How can L'Hôpital's Rule be applied to evaluate lim x→0 (e^x - 1 - x)/x^2, and what does it reveal about the function's behavior?
  • To apply L'Hôpital's Rule to lim x→0 (e^x - 1 - x)/x^2, you differentiate the numerator and denominator because direct substitution gives an indeterminate form. After differentiating twice, you find that the limit evaluates to 1/2. This result indicates that as x approaches 0, the expression (e^x - 1 - x) grows at a rate that is half of x^2.
• Why is it important to recognize that lim x→0 (e^x - 1 - x)/x^2 results in an indeterminate form, and how does this understanding influence your approach to limits?
  • Recognizing that lim x→0 (e^x - 1 - x)/x^2 results in an indeterminate form (0/0) signals that additional techniques like L'Hôpital's Rule must be employed. This understanding shifts your focus from simple substitution to finding derivatives, which allows for a clearer evaluation of limits that might initially seem straightforward. It also highlights the need for deeper analytical skills in calculus.
• Evaluate how the Taylor series expansion for e^x connects with the limit lim x→0 (e^x - 1 - x)/x^2 and explain its significance in understanding exponential functions.
  • The Taylor series expansion for e^x around x = 0 is given by e^x = 1 + x + x²/2 + x³/6 + ..., which directly influences the limit lim x→0 (e^x - 1 - x)/x². By substituting this expansion into the limit, you can see how higher-order terms beyond linear impact the growth rate near zero. The resulting limit being 1/2 illustrates that even though e^x closely resembles linear behavior near zero, it deviates due to quadratic effects, enriching your understanding of exponential functions and their approximation.

Related terms

L'Hôpital's Rule:

A method used to evaluate limits of indeterminate forms by taking derivatives of the numerator and denominator.

Indeterminate Form: A type of limit that does not have a defined value until further analysis is performed, such as 0/0 or ∞/∞.

Taylor Series: An infinite series that represents a function as a sum of terms calculated from the values of its derivatives at a single point.