Limit as x approaches zero of (sinx-(x^3/6)/x^5)
I assume you forgot the parentheses in the numerator,
Lim (sin(x)-(x^3/6))/x^5 as x->0
It would be easy to change sin(x) to a polynomial, the limits of which are easy to find.
If you expand sin(x) by Taylor's series, you'd get
sin(x)=x-x³/6+x^5/120...
So the expression becomes
(x-2x³/6+x^5/120-x^7/5040+...)/x^5 as x->0
which gives +∞
However, if the question had been
Lim (sin(x)-(x-x³/6))/x^5, the result would be 1/120
To find the limit of the given expression as x approaches zero, we can simplify it step by step. Let's start by simplifying the expression:
(sin(x) - x^3/6) / x^5
Next, we can split the expression into two terms:
sin(x)/x^5 - (x^3/6) / x^5
Now, let's simplify each term separately:
Term 1: sin(x)/x^5
As x approaches zero, sin(x) also approaches zero. Additionally, x^5 becomes very small as x approaches zero. Therefore, the limit of this term is 0.
Term 2: (x^3/6) / x^5
By dividing the numerator and the denominator by the highest power of x, which is x^5, we get:
(x^3/6) / x^5 = (1/6) / x^2
As x approaches zero, the denominator x^2 becomes very small, and the term (1/6) / x^2 approaches infinity.
By adding the limits of the two terms, we get:
0 + infinity = infinity
So, the limit of the given expression as x approaches zero is infinity.
To find the limit of the given expression as x approaches zero, we can simplify it using the properties of limits and basic trigonometric identities.
Let's start by simplifying the expression step by step:
1. Evaluate the limit of sin(x) as x approaches zero:
Since sin(x) is a well-known trigonometric function, we know that the limit of sin(x) as x approaches zero is equal to zero itself.
Therefore, the first part of the expression simplifies to zero.
2. Next, simplify the remaining part of the expression:
We have (x^3/6)/x^5.
To simplify this, we divide the numerator and the denominator separately by x^3:
(x^3/6)/(x^5/x^3) = (x^3/6)*(x^3/x^5) = (x^6)/(6x^5) = x/6.
So, the second part of the expression simplifies to x/6.
3. Now, we have the simplified expression:
0 - (x/6) = -x/6.
4. Finally, evaluate the limit of -x/6 as x approaches zero:
Since -x/6 is a linear function, its limit as x approaches zero is simply the function value at zero.
Therefore, the limit of the given expression as x approaches zero is:
lim(x -> 0) (-x/6) = 0/6 = 0.
Hence, the limit as x approaches zero of (sinx-(x^3/6)/x^5) is 0.