The Limit of sinx/x as x Approaches 0 | Understanding Trigonometric Limits in Mathematics

lim x->0 sinx/x

To evaluate the limit of sinx/x as x approaches 0, we can use the concept of a trigonometric limit

To evaluate the limit of sinx/x as x approaches 0, we can use the concept of a trigonometric limit.

Let’s consider the function f(x) = sin(x)/x. When we plug in x = 0 directly into the function, we get an indeterminate form of 0/0, which cannot be evaluated directly.

To overcome this limitation, we can use a trigonometric identity. The limit we want to evaluate can be written as:

lim x->0 sin(x)/x = lim x->0 sin(x)/x * (1/sin(x))

Using the trigonometric identity sinθ/θ = 1 for θ ≠ 0, we can substitute θ = x:

lim x->0 sin(x)/x = lim x->0 1/sin(x)

Now, let’s analyze the limit of 1/sin(x) as x approaches 0. As x approaches 0, sin(x) also approaches 0. We know that sin(0) = 0. Therefore, 1/sin(x) diverges as x approaches 0, meaning it does not have a finite limit.

Hence, the limit of sinx/x as x approaches 0 is undefined.

Note: This result is often referred to as an “undefined limit” or “indeterminate form” since it does not yield a specific numeric value.

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