Solving the Limit of Tangent Function at h=0

In summary, to solve lim x->0 (tan 3(x+h)-tan(3x))/h, you can use the formula for tan(A+B) to simplify tan(3x+3h), then factor out common terms to cancel them out. Additionally, you can use the identity tan(x)/x = 1 to simplify the expression further. This method does not require using trigonometric identities.
  • #1
chukie
80
0
Solve:
lim x->0 (tan 3(x+h)-tan(3x))/h

i hv no clue where to start =(
 
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  • #2
err.. tan (3h)/h?

simply plugging in 0 for x..

edit: are you sure x is approaching 0?

initially, I thought it's h..

then the answer would have been 3+3tan(3x)^2
and you had to do some mess with identites..
http://www.clarku.edu/~djoyce/trig/identities.html
 
Last edited:
  • #3
Did you mean lim h->0??
 
  • #4
Dick said:
Did you mean lim h->0??

sry, yes i mean lim h->0
 
  • #5
Look up a formula for tan(A+B) and apply it to tan(3x+3h). Then simplify, take your limit and use some trig.
 
  • #6
It is actually very simple.. don't even need to any trig after simplifying tan(3x+3h) ..

should factor out things.. and they would cancel out nicely.

And, one more thing tan(x)/x = 1 .. (which is simple to prove is you know sin(x)/x =1 as x-->0)
 
  • #7
rootX said:
It is actually very simple.. don't even need to any trig after simplifying tan(3x+3h) ..

should factor out things.. and they would cancel out nicely.

And, one more thing tan(x)/x = 1 .. (which is simple to prove is you know sin(x)/x =1 as x-->0)

Yep. You don't have to use any trig. But using sec^2(A)=1+tan^2(A) would put it in the simpler form listed in books.
 
  • #8
kk thanks i got it =)
 

1. What is the limit of the tangent function at h=0?

The limit of the tangent function at h=0 is undefined or does not exist.

2. Why is the limit of the tangent function at h=0 undefined?

The tangent function is a periodic function with a period of π, meaning that at h=0, the value of the function jumps from -∞ to ∞. This causes the limit to not exist.

3. Can the limit of the tangent function at h=0 be solved using algebraic methods?

No, the limit of the tangent function at h=0 cannot be solved using algebraic methods. It can only be determined by graphing or using trigonometric identities.

4. How is the limit of the tangent function at h=0 related to the concept of asymptotes?

The limit of the tangent function at h=0 is related to the concept of vertical asymptotes. As h approaches 0, the tangent function approaches vertical lines at x=π/2 and x=-π/2, which are the vertical asymptotes of this function.

5. Are there any real-life applications of solving the limit of the tangent function at h=0?

Yes, the concept of limits and asymptotes, including the limit of the tangent function at h=0, is used in various fields of science and engineering, such as calculus, physics, and electrical engineering. It is also used in computer programming to create smooth curves and avoid errors in calculations.

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