Calculating the limits without the L'Hospital rule

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SUMMARY

The discussion focuses on calculating limits without employing L'Hospital's rule, specifically the limits $$\lim_{x\rightarrow 0}\frac{\sin (x)-x+x^3}{x^3}$$ and $$\lim_{x\rightarrow 0}\frac{e^x-\sin (x)-1}{x^2}$$. Participants confirm that using Taylor series expansions for sine and exponential functions is a valid approach. Additionally, the squeeze theorem is suggested as an alternative method for evaluating these limits.

PREREQUISITES
  • Understanding of Taylor series expansions for sine and exponential functions
  • Familiarity with limit calculations in calculus
  • Knowledge of the squeeze theorem in mathematical analysis
  • Basic proficiency in handling indeterminate forms
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  • Study Taylor series expansions for common functions like sine and exponential
  • Learn how to apply the squeeze theorem in limit problems
  • Explore other methods for evaluating limits, such as series expansion techniques
  • Practice calculating limits without L'Hospital's rule using various approaches
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mathmari
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Hey! :o

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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mathmari said:
Hey! :o

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.
 
Ah ok... Couldn't we use for example the squeeze theorem? (Wondering)
 

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