Lim x→pi/6 e^(sinx)?

Question

Please show work if possible or explain...

Answer 1

This equation is continous everywhere, so all you have to do to evaluate the limit is substitute, plug in Pi/6 for x

sin(pi/6) = 1/2
e^(1/2) = 1.64872127

Answer 2

Since pi(radians)=180 (degrees) pi(radians)/6=30 (degrees).

This defines the classical 30-60-90 degree unit triangle with sides of 1 (hypotenuse - radius on the unit circle), sqrt(3)/2 (length - cos) and 1/2 (height - sin).

Since both sinx and e^x are everywhere continuous,

lim x→pi/6 e^(sinx) = e^[lim x→pi/6 (sinx)]

Thus, from both directions,

lim x→pi/6 e^(sinx) = e^(1/2)

Answer 3

This is just a substitution technique! just substitute pi/6 to x in e^(sinx)

thus, Lim x-> pi/6 e^(sinx) = e^1/2.

This is actually by a theorem of continuity that states that if f is continuous, lim f(g(x)) = f(lim g(x)). i.e. the continuous function and the limit operator are interchangeable.

Thus, rigorously applying the theorem, since the exponential and trigonometric function are continuous at pi/6 (actually, everywhere), then

lim x->pi/6 exp(sin(x)) = exp(lim x->pi/6 (sin x))

= exp ( sin (lim x->pi/6 x)) = exp(sin pi/6) = exp (1/2).

Answer 4

Lim x→pi/6 e^(sinx)?
sin (pi/6) = .5
e^.5 = 1.6487
Lim x--> pi/6 (e^sinx) = e^.5 = approx = 1.6487

Answer 5

omg thats so easy, Lim x→pi/6 e^(sinx)=52