lim tan x - x/
x -> 0 x^2
Lim (tanx -x)/x^2 ?
x->0
The limit of the numerator is zero and so is that of the denominator. Use L'Hopital's rule twice
Lim (sec^2x -1)/2x =
x->0
Lim (2 secx*tanx*secx)/2 =
x->0
= 0
To find the limit of the expression (tan x - x) / x^2 as x approaches 0, you can use algebraic simplification and some trigonometric properties. Here's how you can do it step by step:
Step 1: Simplify the expression:
Divide both the numerator and the denominator by x.
(tan x - x) / x^2 = (tan x / x^2) - (x / x^2) = (tan x / x^2) - (1 / x)
Step 2: Simplify the term tan x / x^2:
As x approaches 0, both x and x^2 approach 0, and we need to determine the behavior of tan x as x approaches 0.
Using the Maclaurin series expansion for tan x, we have:
tan x = x + (x^3 / 3) + (2x^5 / 15) + ...
Dividing both sides by x^2, we get:
tan x / x^2 = (x / x^2) + (x^3 / 3x^2) + (2x^5 / 15x^2) + ...
As x approaches 0, all the terms in the series become 0 except for the first term x / x^2, which simplifies to 1 / x. Therefore, tan x / x^2 approaches 1 / x as x approaches 0.
Step 3: Simplify the term x / x^2:
x / x^2 = 1 / x
Step 4: Substitute the simplified terms back into the original expression:
(tan x - x) / x^2 = (1 / x) - (1 / x)
Step 5: Combine like terms:
(1 / x) - (1 / x) = 0
Therefore, the limit of (tan x - x) / x^2 as x approaches 0 is 0.