Limit WITHOUT using l'hospital

• Doom of Doom
In summary, a limit is the value that a function or sequence approaches as the input or index approaches a certain value. To calculate a limit without using l'Hospital's rule, one can use algebraic manipulation, substitution, or the limit laws. Common techniques for evaluating limits include direct substitution, factoring, rationalization, trigonometric identities, and the squeeze theorem. Understanding limits without using l'Hospital's rule allows for a deeper understanding of function and sequence behavior and serves as a foundation for advanced calculus concepts. However, there are limitations to using l'Hospital's rule, such as only being applicable to certain indeterminate forms and potentially providing less accurate answers. It is important to understand other techniques for evaluating limits to ensure accuracy.
Doom of Doom
For $$p,q\in \mathbb{N}$$, determine $$\lim_{x\to1} \frac{x^{p}-1}{x^{q}-1}$$

I know that it should be p/q. Without using L'hospitals rule, how do I determine it?

The quotient can be rewritten as (1-xp )/(1-xq ). Since x is less than 1 try a binomial expansion.

Factoring :)

NoMoreExams said:
Factoring :)

Got it! My methods of trying were way more complicated. Don't know why I didn't think of that at first!

(x^n) - 1=(x-1)(x^(n-1) + ... + x + 1)

Thanks NoMoreExams

1. What is the definition of a limit?

A limit is the value that a function or sequence approaches as the input or index approaches a certain value.

2. How do you calculate a limit without using l'Hospital's rule?

To calculate a limit without using l'Hospital's rule, you can use algebraic manipulation, substitution, or the limit laws to simplify the function and then evaluate the limit.

3. What are the common techniques to evaluate limits?

The common techniques to evaluate limits include direct substitution, factoring, rationalization, trigonometric identities, and the squeeze theorem.

4. Why is it important to understand limits without using l'Hospital's rule?

Understanding limits without using l'Hospital's rule allows for a deeper understanding of the behavior of functions and sequences and provides a foundation for more advanced calculus concepts.

5. Are there any limitations to using l'Hospital's rule for evaluating limits?

Yes, l'Hospital's rule can only be used for certain types of indeterminate forms, such as 0/0 or ∞/∞. Additionally, it may not always provide the most accurate answer and may require multiple applications to find the limit. It is important to understand other techniques for evaluating limits to ensure accuracy.

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