lim x-> 0 if tan x - x / x^2
Why are you posting this a second time? I answered it earlier thig morning.
And don't you mean Linit "of" instead of "if" ?
http://www.jiskha.com/display.cgi?id=1281878476
To find the limit of the given expression, we will apply algebraic manipulation and some trigonometric identities.
First, let's simplify the expression by factoring out an 'x' from both terms in the numerator:
lim x->0 ((tan x - x) / x^2)
Next, we can use a well-known trigonometric identity: lim x->0 (tan x / x) = 1. Applying this identity, we have:
lim x->0 ((tan x - x) / x^2)
= lim x->0 (tan x / x^2 - x / x^2)
= lim x->0 (tan x / x^2) - lim x->0 (x / x^2)
= (lim x->0 (tan x / x)) / (lim x->0 (x^2))
Now, let's evaluate each of these limits separately.
First, lim x->0 (tan x / x):
To evaluate this limit, we can use the fact that lim x->0 (tan x / x) = 1 as mentioned earlier.
Therefore, lim x->0 (tan x / x) = 1.
Now, let's evaluate the second limit, lim x->0 (x^2):
Since we have a polynomial, the limit is straightforward to find. We substitute the value of 'x' into the expression:
lim x->0 (x^2) = (0)^2 = 0
Substituting the values back into the original expression:
= (lim x->0 (tan x / x)) / (lim x->0 (x^2))
= (1 / 0)
Here, we have an indeterminate form of "1/0", which represents an undefined value. This means that the limit does not exist.
Hence, the limit of the expression as x approaches 0 is undefined.