How can I solve this limit without using d'hospital method?

  • Thread starter Thread starter player1_1_1
  • Start date Start date
player1_1_1
Messages
112
Reaction score
0

Homework Statement


hello, what can I do to solve this without using d'hospital method?
\lim_{x\to0}\frac{\tan x-x}{x^3}
 
Physics news on Phys.org
player1_1_1 said:

Homework Statement


hello, what can I do to solve this without using d'hospital method?
\lim_{x\to0}\frac{\tan x-x}{x^3}

Recall the formula [which is just the M.S. for tan(x)]:
tan(x)=x+\frac{1}{3}x^3+\frac{2}{15}x^5+\frac{17}{315}x^7+...
Substitute this in the limit and the two "x" will be cancelled.
Then divide the denominator and the numerator by x^3.
and the limit will be done by the direct substitution.
The limit = 1/3
 
look up the taylor series for sin and cosine and write the one for tan. Now factor out X^3 from the top and take that limit as X goes to Zero
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
View full post »

Similar threads

Limit of a Function at a Specific Value
Replies
10
Views
2K
Limit problem involving two circles and a line
Replies
8
Views
1K
Limit of piecewise function using epsilon delta
Replies
31
Views
1K
Problems solving a limit which results in an indeterminate form
Replies
4
Views
2K
Uniform Continuity of functions
Replies
40
Views
4K
Evaluate the limit of this harmonic series as n tends to infinity
Replies
17
Views
2K
Solving system of differential equations using elimination method
Replies
10
Views
1K
Solving a Limit Problem: Step-by-Step Guide
Replies
3
Views
2K
Proving piecewise function is k-differentiable
Replies
5
Views
2K
Find limit of multi variable function
Replies
10
Views
1K

Hot Threads

Recent Insights

Back
Top