lim x->0 sinbx/x
b
To evaluate this limit, we can use the L’Hopital’s rule. However, before we do that, we need to simplify the expression by multiplying and dividing by the conjugate:
lim x->0 [(sin bx/x) * (bx)/(bx)]
Now, the numerator and denominator are the same, so we can rewrite the expression as:
lim x->0 [(sin bx – bx)/x * -(1/bx)]
Applying L’Hopital’s rule, we take the derivative of the numerator and denominator separately:
lim x->0 [(b cos bx – b)/1 * -(1/bx)]
= lim x->0 [(cos bx – 1)/x * -1]
= -b lim x->0 [(sin bx)/bx * 1/x]
= -b
Therefore, the limit of sinbx/x as x approaches 0 is equal to -b.
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