Evaluate the limit as x approaches 0 of [tan(x+Δx)-tan(x)] /Δx:
sec^ 2 (x)
cot (x)
sec (x)
does not exist
This is handily explained at
http://answers.yahoo.com/question/index?qid=20100519122921AAIzxX9
To evaluate the limit as x approaches 0 of [tan(x+Δx)-tan(x)] /Δx, we first simplify the expression:
[tan(x+Δx)-tan(x)] /Δx
Next, we use the trigonometric identity for the difference of two tangents to simplify further:
tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))
Applying this identity, we have:
[tan(x+Δx)-tan(x)] /Δx = [tan((x + Δx) - x)] / (1 + tan(x)(tan(x + Δx)))
Simplifying, we get:
[tan(Δx)] / (1 + tan(x)(tan(x + Δx)))
Now, as x approaches 0, the value of tan(x) approaches 0, and we can make the approximation tan(x) ≈ x for small x. So, tan(x)(tan(x + Δx)) becomes x(x + Δx), which simplifies to x^2 + xΔx.
Substituting this approximation into our expression, we get:
[tan(Δx)] / (1 + x(x^2 + xΔx))
Now, as Δx approaches 0, tan(Δx) approaches 0, and the denominator remains constant. Thus, the limit as x approaches 0 of the given expression is:
lim Δx→0 [tan(Δx)] / (1 + x(x^2 + xΔx))
= 0 / (1 + x(x^2))
= 0
Therefore, the limit as x approaches 0 of [tan(x+Δx)-tan(x)] /Δx is 0.
The answer is not cot(x), sec^ 2 (x), sec(x), or does not exist. It is 0.