Calculating the Limit as x Approaches 0 of (1 – cosx)/x Using Trigonometric Identities

limit as x approaches 0: 1-cosx/x

(1 – cosx)/x is sin(x/2)

To find the limit as x approaches 0 of the expression (1 – cosx)/x, we can use algebraic manipulation and trigonometric identities.

Step 1: Simplify the expression by factoring out a -1 from the numerator:
(1 – cosx)/x = – (cosx – 1)/x

Step 2: Rewrite cosx – 1 as -2sin^2(x/2) using the double angle identity for cosine:
– (cosx – 1)/x = 2sin^2(x/2)/x

Step 3: Divide both the numerator and denominator by x:
2sin^2(x/2)/x = 2(sin(x/2)/x)(sin(x/2))

Step 4: Apply the limit properties:
As x approaches 0, sin(x/2)/x approaches 1/2. So the expression becomes:
2(1/2)(sin(x/2)) = sin(x/2)

Therefore, the limit as x approaches 0 of (1 – cosx)/x is sin(x/2).

Definitions:

1. Limit: In mathematics, the limit of a function is the value that the function approaches as the input (x) gets arbitrarily close to a certain value. It can be denoted using limit notation, such as lim(x→a) f(x), where “a” is the value that x approaches.

2. Trigonometric Functions: These are mathematical functions related to angles and circles. The most common trigonometric functions include sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These functions have various properties and relationships, and are widely used in fields such as geometry, physics, and engineering.

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