Solving the Limit of sin(bx)/x using L’Hôpital’s Rule: Explanation and Answer

lim x->0 sinbx/x

To evaluate the limit as x approaches 0 of sin(bx)/x, where b is a constant, we can use L’Hôpital’s Rule

To evaluate the limit as x approaches 0 of sin(bx)/x, where b is a constant, we can use L’Hôpital’s Rule.

Applying L’Hôpital’s Rule, take the derivative of the numerator and denominator separately. The derivative of sin(bx) with respect to x is b*cos(bx), and the derivative of x with respect to x is 1.

Therefore, the limit becomes:

lim x->0 (b*cos(bx))/1

Now, we can substitute x = 0 into the expression:

lim x->0 (b*cos(0))/1

cos(0) is equal to 1, so the expression simplifies to:

lim x->0 b/1

Since x is approaching 0, we can ignore it in the limit. Thus, the final limit is:

b

Therefore, lim x->0 sin(bx)/x = b.

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