Why Sinx by x = 1? Differencial Calculus. Limit f sinx/x
For class 12, Class 11, JEE MAIN, NEET students.
Hello friends!
Today, we are going to realize, the limiting value of sine x, divided by, x, which is vastly used in study of calculus.
Suppose, O A D, is a right angled triangle, whose Angle, A O D, has a value of, x. Where, x is measured in radian. And, its Hypotenuse O A, has a fixed length of r. Angle, A O D, can change its value, freely. So, the other two arms of Triangle, namely, the arm O D, and the arm O A, undergo a variation of lengths. From the principle, sine x is, A D, divided by, r.
Note that, values of sine x is smaller, when x is smaller.
Also, we have a variable angle with us. Its name is, angle, A O B. Its value is, x, and, measured in radian. If, the radius of the Circle is, r. Then, we get, x equals, arc A B, divided by, r.
Now, superpose these two diagrams. And, observe the variations.
Also, take ratio of, sine x, and, x. Thus, sine x, divided by x, equals, arm A D, divided by arc A B. So, the value of the ratio, sine x divided by x, depends on two quantities. First one, is the length of the arm, A D. And, the second one, is the length of the arc, A B.
NOW, note the variations. If, x is about one radian, then arc A B is much longer than arm A D. As x tends to zero, the arm A D, and the arc A B, become equal in length. Thus, the limiting value of sine x divided by x is, 1.
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