How to Evaluate the Limit of the Function f(x) = sin(bx)/x as x Approaches 0

lim x->0 sinbx/x

To evaluate the limit of the function f(x) = sin(bx)/x as x approaches 0, we can use L’Hôpital’s rule or the concept of the derivative if b ≠ 0

To evaluate the limit of the function f(x) = sin(bx)/x as x approaches 0, we can use L’Hôpital’s rule or the concept of the derivative if b ≠ 0.

Using L’Hôpital’s rule:

Step 1: Take the derivative of the numerator and denominator separately.
f'(x) = (d/dx)(sin(bx)) / (d/dx)(x)
= bcos(bx) / 1
= bcos(bx)

Step 2: Replace x with 0 in the derivative obtained:
lim x->0 bcos(bx) = bcos(0) = b

Using the concept of the derivative:

Let y = bx, then x = y/b. The expression becomes f(y) = sin(y/b)/(y/b).

Taking the derivative of the new function with respect to y:
f'(y) = (d/dy)(sin(y/b))/(d/dy)(y/b)
= (1/b)cos(y/b)/(1/b)
= cos(y/b)

Taking the limit as y approaches 0:
lim y->0 cos(y/b) = cos(0/b) = cos(0) = 1

Step 3: Replace y with 0 in the derivative obtained:
lim y->0 cos(y/b) = cos(0/b) = cos(0) = 1

Therefore, the limit of f(x) = sin(bx)/x as x approaches 0 is equal to b if b ≠ 0, or 1 if b = 0.

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