Find the limit
lim as x approaches (pi/2) e^(tanx)
I have the answer to be zero:
t = tanx
lim as t approaches negative infi e^t
= 0
Why is tan (pi/2) approaching negative infinity is my question?
tan x = sin x / cos x
as x-->pi/2
sin x --> +1 and cos x-->0
cos goes to +0 from the first quadrant and cos goes to -0 from the second quadrant as x-->pi/2
so depending on if you approach pi/2 from right or from left, tan x -->__+oo or -oo
To understand why tan(pi/2) approaches negative infinity, we need to look at the graph of the tangent function.
The tangent function is defined as tan(x) = sin(x) / cos(x). At x = pi/2 (or 90 degrees), the cosine function becomes zero (cos(pi/2) = 0).
Now let's examine the behavior of the tangent function as x approaches pi/2 from the left and right sides.
As x approaches pi/2 from the left side (x < pi/2), the sine function remains positive, and the cosine function becomes very close to zero. When we divide a positive number by a very small positive number, we get a large positive number. Therefore, the tangent function approaches positive infinity as x approaches pi/2 from the left side.
As x approaches pi/2 from the right side (x > pi/2), the sine function remains positive, but the cosine function becomes very close to zero. Now, when we divide a positive number by a very small negative number (since cos(x) is negative for x > pi/2), we get a large negative number. In this case, the tangent function approaches negative infinity as x approaches pi/2 from the right side.
Since the limit we are evaluating is as x approaches pi/2, it includes both the left and right side behavior. Therefore, we can say that tan(pi/2) approaches negative infinity.
In the calculation you provided, you substitute t = tan(x), and as t approaches negative infinity, the limit of e^t approaches zero.
Let's analyze why tan(pi/2) approaches negative infinity.
In trigonometry, the tangent function is defined as the ratio of the sine of an angle to the cosine of the same angle. Mathematically, tan(x) = sin(x)/cos(x).
At pi/2, the cosine of this angle is 0. Recall that the cosine function is equal to 0 at pi/2 and its multiples. Thus, cos(pi/2) = 0.
Since division by zero is undefined, tan(pi/2) is infinite or undefined.
However, we need to determine whether tan(pi/2) approaches positive or negative infinity. To do this, we look at the behavior of the tangent function near pi/2.
By analyzing smaller values approaching pi/2, we can see that the values of tan(x) become larger and larger in magnitude as x approaches pi/2 from the left side (x < pi/2). This means that the limit of tan(x) as x approaches pi/2 from the left side is negative infinity, i.e., lim(x --> pi/2-) tan(x) = -∞.
Hence, it is correct to say that tan(pi/2) approaches negative infinity.