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Sin 1x1 [UPDATED]


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sin 1x1



A zero-bounded limit is one in which the function can be brokeninto a product of two functions where one function converges to zero and the otherfunction is bounded. If we show that a limit is zero -bounded, then the zero-boundedlimit theorem implies that the limit goes to zero.


Instead of looking at $\frac\sin(1/x)1/x$ for small $x$, you can look at $\frac\sin(y)y$ for large $y$. Now things should be more obvious: the denominator gets large, while the absolute value of the numerator stays less then or equal to $1$. Hence, the fraction will tend to $0$.


Now, the derivative of sin inverse x can be calculated using different methods. It can be derived using the limits definition and inverse function theorem. In this article, we will calculate the derivative of sin inverse x and also discuss the anti-derivative of sin inverse x which is nothing but the integral of sin-1x.


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By differentiating we obtain $$f'(x)=f(x+1)-f(x-1)$$This type of equations was addressed in the MO questionOn equation f(z+1)-f(z)=f'(z)Let $\lambda$ be any (complex) root of the equation$$\lambda=e^\lambda-e^-\lambda,$$which is equivalent to $z=2\sin z$, as Noam wrote.to this $\lambda$ a solution $f(x)=e^\lambda x$ is associated.So you have infinitely many exponential solutions. Any linear combination isalso a solution. Then, depending on your assumptions of $f$ you can consider various limits of those linear combinations in the appropriate topology for yourfunctions/distributions class.


Edit. For a complete theory of this kind of equations see "Fonctions moyenne-periodiques", a theory created by Delsart and Schwartz in 1940-s.This is generalized in the modern theory of "equations of convolution-type", see,for example Hormander, Linear Partial differential operators. In this theory one considers equations $u\star f=0$, where $u$ is a distribution. In your case,$u=\delta-\chi,$ where $\chi$ is the characteristic function of $[-1,1]$. The method of solution is an appropriate version of Fourier--Laplace transform, depending on your class of functions/distributions. Ordinary Fourier transform of $u$ is $U(\lambda)=1-2\sin\lambda/\lambda$, whose roots give you the exponential solutions.


When you use sym on a numeric input, the numeric expression is first evaluated to the MATLAB default double-precision number that can be less accurate. Then, sym is applied on that double-precision number. To represent an exact number without evaluating it to double precision, use a character vector with quotes. For example, create a symbolic number to represent a very large integer exactly.


When you use vpa on a numeric expression, such as log(2), the expression is first evaluated to the MATLAB default double-precision number that has less than 32 significant digits. Then, vpa is applied on that double-precision number, which can be less accurate. For more accurate results, convert double-precision numbers in an expression to symbolic numbers with sym and then use vpa to evaluate the results with variable precision. For example, find log(2) with 17- and 20- digit precision.


The syms command is shorthand for the sym syntax, but the two functions handle assumptions differently. syms clears the assumptions when creating variables. However, recreating a variable using sym does not clear its assumptions. For more details about the differences of these two functions, see Choose syms or sym Function.


The syms command is a convenient shorthand for the sym syntax, and its typical use is to create fresh symbolic variables for interactive symbolic workflows. Use the sym syntax to create the following:


You can also apply a mathematical function to an arithmetical expression. For example, apply the Bessel function of the first kind J0 to the arithmetical expression f and find its derivative with respect to x.


To create a symbolic number in a symbolic expression, use sym. Do not use syms to create a symbolic expression that is a constant. For example, to create an expression whose value is 5, enter f = sym(5). The command f = 5 does not define f as a symbolic expression.


You can use the syms command to clear variables of definitions that you previously assigned to them in your MATLAB session. syms clears the assumptions of the variables. These assumptions (which can be real, integer, rational, and positive) are stored separately from the symbolic object. However, recreating a variable using sym does not clear its assumptions. For more information, see Delete Symbolic Objects and Their Assumptions.


Mistake Points[-2, 0] is different from [-2, 0). '[' and ']' indicates that the end number (2 and 0) is also included. '(' and ')' indicates that 2 and 0 are not taken into consideration.


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