lim (4sin2x - 3x cos5x)/(3x/2 +(x^2)cscx )
x->0
L'Hoptial's rule applies to this.
I get for the numerator..
8cos2x - 3cos5x - 15x sin5x
and the denominator...
3/2 + 2x csc x + x^2 d cscx/dx
so the limit looks to be
5/ (3/2) = 10/3
check me.
To find the limit of the given expression as x approaches 0, you can use L'Hôpital's rule, which states that if the limit of the ratio of two functions as x approaches a particular value is in the form of 0/0 or ∞/∞, then the limit of the ratio is equal to the limit of the ratio of their derivatives.
Let's apply L'Hôpital's rule to the given expression step by step.
Step 1: Find the derivatives of the numerator and the denominator separately.
Numerator:
Take the derivative of each term using the chain rule and product rule.
d/dx (4sin^2x - 3x cos^5x)
= 8sinxcosx - 3cos^5x + 15x^4sinxsin^4x
Denominator:
Take the derivative of each term using basic derivative rules.
d/dx (3x/2 +(x^2)cscx)
= 3/2 + 2xcscx - x^2cscxcotx
Step 2: Simplify the derivatives obtained in step 1.
Numerator:
Simplify the expression obtained for the derivative of the numerator, if possible.
d/dx (4sin^2x - 3x cos^5x)
= 8sinxcosx - 3cos^5x + 15x^4sinxsin^4x
Denominator:
Simplify the expression obtained for the derivative of the denominator, if possible.
d/dx (3x/2 +(x^2)cscx)
= 3/2 + 2xcscx - x^2cscxcotx
Step 3: Take the limit of the ratio of the derivatives as x approaches 0.
lim (8sinxcosx - 3cos^5x + 15x^4sinxsin^4x) / (3/2 + 2xcscx - x^2cscxcotx)
x->0
Now, substitute x=0 into the simplified derivatives.
lim (8sin(0)cos(0) - 3cos^5(0) + 15(0)^4sin(0)sin^4(0)) / (3/2 + 2(0)csc(0) - (0)^2csc(0)cot(0))
x->0
Simplifying further:
lim (0 - 3(1) + 0) / (3/2 + 0 + 0)
x->0
lim (-3) / (3/2)
x->0
Finally, simplify the expression:
lim (-3) / (3/2)
= -2
Therefore, the limit of the given expression as x approaches 0 is -2.