The lim as x goes to 0- for cos(x)/sin(x). I know the answer is negative infinity but I don't really understand why.
sin(-0.000000001)=-0.000000001
cos(-0.000000001)=0.999999999
sin/cos=-999999999
I mean
cos(-0.000000001)/sin(-0.000000001)=
=-999999999
To find the limit of cos(x)/sin(x) as x approaches 0 from the left (x -> 0-), we can use the properties of trigonometric functions and their limits.
First, let's recall the definitions of cos(x) and sin(x). Cos(x) represents the ratio of the adjacent side to the hypotenuse in a right triangle, and sin(x) represents the ratio of the opposite side to the hypotenuse.
In this case, as x approaches 0 from the left, x becomes slightly negative but very close to 0. Therefore, we can consider the right triangle in the second quadrant of the unit circle, where both the adjacent and opposite sides are positive.
Now, let's consider the limit expression cos(x)/sin(x).
As x approaches 0 from the left, cos(x) approaches 1, and sin(x) approaches 0, since the value of sin(x) is getting smaller and closer to 0 as x approaches 0.
Since the denominator (sin(x)) approaches 0, and the numerator (cos(x)) is approaching a non-zero value, the quotient goes to infinity or negative infinity depending on the sign.
In this case, since we are approaching from the left side (x -> 0-), the value of sin(x) is positive, and since cos(x) is positive in the second quadrant, the limit cos(x)/sin(x) approaches negative infinity.
Thus, the limit as x approaches 0- for cos(x)/sin(x) is negative infinity.