Limit of e^(2x) / sinh(2x) as x approaches infinity

In summary, the limit of e^(2x) / sinh(2x), as x approaches infinity, can be simplified to ( ∞ ) / [( ∞ - 0 ) / 2]. Dividing top and bottom by e2x results in 2 / (1 - e^(-4x)), which approaches 2 as x approaches infinity. Therefore, the overall limit is 2.
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Homework Statement



Find the limit of e^(2x) / sinh(2x), as x approaches infinity.


Homework Equations





The Attempt at a Solution



Changed equation to e^(2x) / [(e^(2x) - e^(-2x)) / 2]. Based of the sinh identity.

found limit of e^(2x) = ∞ ... e^(-2x) = 0 ... 2 = 2 ...

New equation ( ∞ ) / [( ∞ - 0 ) / 2]

...

I am unsure how to continue after this.
 
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Welcome to PF!

Hi ccha ! Welcome to PF! :smile:

(try using the X2 tag just above the Reply box :wink:)
ccha said:
… Changed equation to e^(2x) / [(e^(2x) - e^(-2x)) / 2]. Based of the sinh identity.

Now divide top and bottom by e2x

what do you get? :smile:
 

What is the definition of a limit of a hyperbolic?

The limit of a hyperbolic function is the value that the function approaches as the independent variable approaches a specific value. It is denoted by the symbol "lim" and is used to describe the behavior of a function near a specific point.

How is the limit of a hyperbolic calculated?

The limit of a hyperbolic function is calculated by plugging in the specific value that the independent variable is approaching into the function and evaluating the resulting expression. This value will represent the limit of the function at that particular point.

Can a hyperbolic function have multiple limits?

Yes, a hyperbolic function can have multiple limits. These limits can vary depending on the direction in which the independent variable is approaching the specific value. If the limit from the left and the limit from the right are equal, then the overall limit exists. Otherwise, the limit does not exist.

What is the difference between a finite and infinite limit of a hyperbolic?

A finite limit of a hyperbolic is when the function approaches a specific value as the independent variable gets closer to a particular point. An infinite limit is when the function approaches either positive or negative infinity as the independent variable gets closer to a specific point.

What is the significance of the limit of a hyperbolic in mathematics and real-world applications?

The limit of a hyperbolic is an essential concept in mathematics as it helps us understand the behavior of functions near specific points. This concept is also used in real-world applications, such as in physics, engineering, and economics, to model and analyze various phenomena.

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