Calculating the limits without the L'Hospital rule

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In summary, the conversation is about calculating limits without using the L'Hospital rule. The question is whether the Taylor expansion is the only way to do so, and the other person suggests using the squeeze theorem.
  • #1
mathmari
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Hey! :eek:

How could we calculate the following limits without the L'Hospital rule?

$$\lim_{x\rightarrow 0}\frac{\sin (x)-x+x^3}{x^3} \\ \lim_{x\rightarrow 0}\frac{e^x-\sin (x)-1}{x^2}$$

Is the only way using the Taylor expansion? (Wondering)
 
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  • #2
mathmari said:
Hey! :eek:

How could we calculate the following limits without the L'Hospital rule?

$$\lim_{x\rightarrow 0}\frac{\sin (x)-x+x^3}{x^3} \\ \lim_{x\rightarrow 0}\frac{e^x-\sin (x)-1}{x^2}$$

Is the only way using the Taylor expansion? (Wondering)
It certainly looks as though they are expecting you to use the Taylor series expansions for the sine and exponential functions.
 
  • #3
Ah ok... Couldn't we use for example the squeeze theorem? (Wondering)
 

1. What is the L'Hospital rule and why can't it be used to calculate limits?

The L'Hospital rule is a mathematical rule used to evaluate limits of indeterminate forms. It states that for certain functions, the limit of their ratio is equal to the limit of their derivatives. However, this rule cannot be applied to all functions and can only be used in specific cases.

2. How do you calculate limits without using the L'Hospital rule?

There are several methods to calculate limits without using the L'Hospital rule, such as factoring, rationalization, substitution, and using basic limit laws. These methods involve manipulating the given function algebraically to simplify it and then evaluating the limit by plugging in the limit value.

3. Can I always calculate limits without using the L'Hospital rule?

No, not all limits can be evaluated without using the L'Hospital rule. Some limits may require advanced techniques such as Taylor series or integration to evaluate them. It is important to understand the limitations and conditions of each method before attempting to calculate a limit.

4. Are there any advantages to calculating limits without using the L'Hospital rule?

Yes, there are several advantages to calculating limits without using the L'Hospital rule. It allows for a better understanding of the fundamental concepts of limits and helps to develop problem-solving skills. Additionally, it can be used in cases where the L'Hospital rule cannot be applied, providing a more comprehensive approach to calculating limits.

5. Can you provide an example of calculating a limit without using the L'Hospital rule?

Sure, let's consider the limit of the function f(x) = (x^2 - 4)/(x - 2) as x approaches 2. Using the L'Hospital rule, we would have to take the derivative of the numerator and denominator separately, resulting in a limit of 4. However, using substitution, we can rewrite the function as f(x) = (x - 2)(x + 2)/(x - 2). Canceling out the common factor of (x - 2), we are left with the limit of f(x) = x + 2. Plugging in the limit value of x = 2, we get a limit of 4, the same result as using the L'Hospital rule.

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