- #1
zeion
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- 1
Homework Statement
[tex]\lim_{x \to 0} \frac{e^x - e^{-x} - 2x}{x - sinx}[/tex]
Homework Equations
The Attempt at a Solution
Do I need to do a l'hopital?
zeion said:What do I do with a [tex]e^{-0}[/tex]? Is that the same as just 0
zeion said:1
Ok so I do some l'hops and I get this
[tex]\frac {e^x + e^{-x} - x}{1 - cosx} = \frac {e^x - e^{-x} - 1}{sinx} = \frac {1-1-1}{1} = -1[/tex]
zeion said:What do I do with a [tex]e^{-0}[/tex]? Is that the same as just 0
Dick said:Sure. -0 is the same as 0. What's e^0?
zeion said:Ok sorry so I get like
[tex]\frac {e^x + e^{-x} -2}{1-cosx} = \frac {e^x - e^{-x}}{sinx} = \frac {e^x + e^{-x}}{cosx} = 2?[/tex]
zeion said:Ok sorry so I get like
[tex]\frac {e^x + e^{-x} -2}{1-cosx} = \frac {e^x - e^{-x}}{sinx} = \frac {e^x + e^{-x}}{cosx} = 2?[/tex]
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a specific value. It is used to determine the value that a function approaches, but may not necessarily reach, as its input approaches a particular point.
To find the limit of a function, first evaluate the function at the input value it is approaching. Then, try to simplify the expression as much as possible. If the simplified expression results in a finite number, then that is the limit of the function. If it is not possible to simplify the expression, you may need to use more advanced techniques such as L'Hopital's rule or the squeeze theorem.
While most calculators have the capability to evaluate limits, it is important to understand the concepts and techniques used to find limits rather than relying solely on technology. Additionally, some limits may not be possible to evaluate using a calculator, so it is important to have a solid understanding of the underlying principles.
Some common techniques for evaluating limits include direct substitution, factoring, rationalization, and trigonometric identities. Additionally, there are more advanced techniques such as L'Hopital's rule, the squeeze theorem, and series expansion that can be used for more complex limits.
Limits are used in many real-world applications, especially in science and engineering. For example, they are used to model the behavior of systems that approach certain values, such as the speed of an object approaching a particular velocity or the concentration of a chemical as it approaches equilibrium. Limits also play a crucial role in understanding the concept of continuity, which is essential in many fields of study.