Find the limit x->pi/2^+ tan(x)
I know the answer is -∞, but I don't understand why. When I simplify to sin(x)/cos(x), I get 1/0^+.
tan=sin(x)/cos(x) is a good start.
How about putting
lim x&rarrow;pi+
as
x&rarrow; as (π/2+δ) as δ-> 0+.
tan(x)
=sin(x)/cos(x)
=sin(π/2+δ)/cos(π/2+δ)
=(sin(π/2)cos(δ)+cos(π/2)sin(δ))/ (cos(π/2)cos(δ)-sin(π/2)sin(δ))
setting sin(π/2)=1 and cos(π/2)=0,
=(cos(δ)+0*sin(δ))/ (0*cos(δ)-sin(δ))
=cos(δ)/(-sin(δ)
and setting cos(δ)=1, sin(δ)=0 as δ->+0
=1/(-0)
=-∞
π/2 is 90º
on the unit circle ... tan = y/x
at 90º, x = 0
dividing by zero gives the infinite limit
depending on the direction of approach to π/2, the sign of the limit can be +/-
To find the limit as x approaches π/2 from the right side of the given function, tan(x), we can simplify it to sin(x)/cos(x), as you mentioned. However, it is important to note that 1/0^+ is not a valid expression.
To understand why the limit is -∞, let's examine the behavior of tan(x) as x approaches π/2 from the right side.
As x approaches π/2 from the right side, the value of cos(x) approaches 0, while sin(x) remains positive. This is because at π/2, the cosine function has a vertical asymptote, and the value approaches zero from positive values.
Since tan(x) is defined as sin(x)/cos(x), as cos(x) approaches 0 from the positive side, the quotient sin(x)/cos(x) will tend towards positive infinity. Therefore, the limit of tan(x) as x approaches π/2 from the right side is +∞.
Now, it might seem contradictory to what you said earlier, but we can explain why the answer is actually -∞. When we write π/2^+, it means we are approaching π/2 from values greater than π/2, but not equal to π/2.
By considering the interval (π/2, π), tan(x) is negative. As we approach π/2 from the right side, which is a little bit smaller than π/2, the value of tan(x) becomes unlimitedly large in the negative direction, reaching negative infinity. Thus, the limit of tan(x) as x approaches π/2 from the right side, more accurately expressed as x->π/2^+, is indeed -∞.
In conclusion, the limit of tan(x) as x approaches π/2 from the right side is -∞.