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How do you calculate the limit $\displaystyle \lim_{x \to \infty}\frac{x}{\cosh{x}}$
Thanks. Is this correct?greg1313 said:\(\displaystyle \cosh(x)=\dfrac{e^x+e^{-x}}{2}\)
Guest said:How do you calculate the limit $\displaystyle \lim_{x \to \infty}\frac{x}{\cosh{x}}$
A hyperbolic limit refers to the limit of a function as the input value approaches infinity or negative infinity, in which case the function behaves like a hyperbola.
To calculate the hyperbolic limit of a function, you can use the L'Hopital's rule, which states that the limit of the quotient of two functions is equal to the limit of their derivatives.
The hyperbolic limit of the function $\frac{x}{\cosh{x}}$ is equal to 1. This can be found by taking the derivative of both the numerator and denominator, which results in the limit becoming $\frac{1}{\sinh{x}}$. As $x$ approaches infinity or negative infinity, $\sinh{x}$ also approaches infinity, resulting in the limit being equal to 1.
For example, let's calculate the limit of the function $\frac{2x}{\cosh{x}}$ as $x$ approaches infinity. We can use L'Hopital's rule and take the derivative of both the numerator and denominator to get $\frac{2}{\sinh{x}}$. As $x$ approaches infinity, $\sinh{x}$ also approaches infinity, resulting in the limit being equal to 0.
Calculating the hyperbolic limit of a function is useful in determining the behavior of the function as the input value approaches infinity or negative infinity. This can help in understanding the overall behavior of the function and making predictions about its values at large inputs.