Simplify, then apply the rules of limit to evaluate
lim x3-5x2+6x/x2-9
x->3
first, we simplify the function:
(x^3-5x^2+6x)/(x^2-9)
x(x^2-5x+6)/(x-3)(x+3)
x(x-2)(x-3)/(x-3)(x+3)
cancel out x-3:
x(x-2)/(x+3)
thus, the limit as x--> is:
3(3-2)/(3+3)
1/2
so there,, :)
Given two functions as:
f(x) = x2-x-1 and g(x) = 3/x
Find fog(x) also find the domain of f, g and fog
Solve the two sided inequality and show the solution on real line
7 < 1-2x ≤ 10
To simplify the expression and apply the rules of limits, we need to factorize the numerator and denominator.
1. Factorize the numerator:
x^3 - 5x^2 + 6x = x(x^2 - 5x + 6)
= x(x - 2)(x - 3)
2. Factorize the denominator:
x^2 - 9 = (x - 3)(x + 3)
After factorizing, we can cancel out the (x - 3) term from the numerator and denominator since the limit is evaluated as x approaches 3.
3. Simplify and cancel out common terms:
lim(x->3) [(x(x - 2)(x - 3))/(x - 3)(x + 3)]
= lim(x->3) [(x - 2)/(x + 3)]
Now, we can directly substitute x = 3 into the simplified expression because (x - 2)/(x + 3) is continuous at x = 3.
4. Substitute x = 3:
= (3 - 2)/(3 + 3)
= 1/6
Therefore, using the rules of limits and the process outlined above, the value of the limit as x approaches 3 is 1/6.