Evaluating the Limit of sin(bx)/x as x Approaches 0 using Trigonometric Identities and the Limit Definition

lim x->0 sinbx/x

To evaluate the limit of sin(bx)/x as x approaches 0, let’s break it down step by step using a trigonometric identity and the limit definition

To evaluate the limit of sin(bx)/x as x approaches 0, let’s break it down step by step using a trigonometric identity and the limit definition.

We start with sin(bx)/x. Since we have x in the denominator, we cannot simply substitute 0 into x directly because it would result in division by zero, which is undefined. To overcome this, we use a trigonometric identity:
lim x->0 sin(bx)/x = lim x->0 (sin(bx)/bx) * (bx/x)

Now, according to the limit properties, we can separate the limits:
lim x->0 sin(bx)/x = lim x->0 sin(bx)/bx * lim x->0 (bx/x)

First, let’s consider the limit lim x->0 (bx/x). Notice that bx is simply b multiplied by x, so we can rewrite this limit as:
lim x->0 (bx/x) = lim x->0 b

Since x is approaching 0, regardless of the value of b, the limit lim x->0 (bx/x) = b.

Now, let’s focus on lim x->0 sin(bx)/bx. According to a well-known trigonometric property:
lim t->0 sin(t)/t = 1

We can apply this to our limit by substituting t with bx:
lim x->0 sin(bx)/bx = 1

Finally, combining the limits we determined:
lim x->0 sin(bx)/x = lim x->0 sin(bx)/bx * lim x->0 (bx/x) = 1 * b = b

Therefore, the limit of sin(bx)/x as x approaches 0 is equal to b.

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