how do you find sin7x/4x when lim x goes to 0
the answer is 7/4
One way would be to write sin 7x in infinite series form, and divide out 4x.
sin7x= 7x - (1/3!)(7x)^3 + (1/5!)(7x)^5 - ...
After dividing by 4x, the only term that does not vanish as x->0 is 7/4.
You could also use L'Hopital's rule. Take the ratio of the x-derivatives of the numerator and denominator.
Using L'Hospital rule
Lim sin7x/4x as x ->0 = Lim sim7x/Lim 4x as x->0 = Sin (Lim 7x)/(Lim 4x)as x-> 0 = 7/4 (on differentiating both numerator and denominator w.r.t. x) = 7/4
To find the limit of sin(7x)/(4x) as x approaches 0, we can follow these steps:
Step 1: Simplify the expression
Since sin(0) = 0, we cannot substitute the limit directly. To simplify the expression, we can use a trigonometric identity:
lim(x->0) sin(kx)/x = lim(x->0) sin(kx)/kx = 1
Applying this identity, we have:
lim(x->0) sin(7x)/(4x) = lim(x->0) (7/4) * (sin(7x)/(7x))
Step 2: Apply the identity
Now, we need to use the trigonometric identity lim(x->0) sin(x)/x = 1. By substituting 7x for x, we get:
7 * lim(x->0) sin(7x)/(7x)
Step 3: Substitute the limit
Using the trigonometric identity mentioned above, we have:
7 * 1 = 7
Therefore, the limit of sin(7x)/(4x) as x approaches 0 is 7/4.