thereddevils said:In one of the examples in my book , it says that
[tex]\lim_{\delta x\rightarrow 0}\frac{\sin \frac{\delta x}{2}}{\frac{\delta x}{2}}=1[/tex]
how can that be ?
Mark44 said:Let's make the limit a little simpler by getting rid of the Greek letters. They don't really add anything and it's a pain to have to type in \delta all the time.
[tex]\lim_{x \rightarrow 0}\frac{\sin \frac{x}{2}}{\frac{x}{2}}=1[/tex]
This can be and is because for values of x that are close to zero, sin(x) is approximately equal to x, making the ratio close to 1. You can convince yourself of the reasonableness of my claim by using your calculator (in radian mode) to calculate sin(x)/x for x = .1, .01, .001, and so on (also for negative x that is close to 0).
In this problem, if x is close to zero, then x/2 will be even closer to zero, so the ratio sin(x/2)/(x/2) will be even closer to 1 than would be the ratio of sin(x)/x.
Note that I am not proving anything here.
thereddevils said:thanks Mark , but why must it be in radian mode ?
The limit of sin(dx/2)/(dx/2) represents the behavior of the function sin(x) as x approaches 0. It is a measure of the rate of change of the sine function at x=0, or the slope of the tangent line at that point.
This limit is a fundamental property of the sine function and is derived from the definition of the derivative. As x approaches 0, the ratio of sin(dx/2) to dx/2 approaches 1, resulting in a limit of 1.
The limit of sin(dx/2)/(dx/2) = 1 is important in calculus because it allows us to calculate the derivatives of trigonometric functions. It is also used in the proof of the derivative of the sine function.
The limit of sin(dx/2)/(dx/2) can also be evaluated using the Maclaurin series expansion of the sine function. This method involves representing the function as an infinite polynomial and taking the limit as the number of terms approaches infinity.
The concept of limits and derivatives is crucial in understanding the behavior of various phenomena in the natural world. For example, the rate of change of a physical quantity, such as velocity or acceleration, can be represented by a derivative, and the limit of sin(dx/2)/(dx/2) = 1 plays a role in calculating these values in certain scenarios.