The Sine Limit: Finding the Limit of sin(bx)/x as x Approaches 0

lim x->0 sinbx/x

To find the limit of sin(bx)/x as x approaches 0, where b is a constant, we can use a common trigonometric limit known as the Sine Limit

To find the limit of sin(bx)/x as x approaches 0, where b is a constant, we can use a common trigonometric limit known as the Sine Limit.

The Sine Limit states that as x approaches 0, the limit of sin(x)/x is equal to 1. This can be proven using the Squeeze Theorem or by using the Taylor series expansion for sin(x).

We can use this Sine Limit to solve the given limit. Let’s substitute bx for x:

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

Since bx approaches 0 as x approaches 0, we can apply the Sine Limit:

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

Thus, the limit of sin(bx)/x as x approaches 0, where b is a constant, is equal to 1.

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