Calculus (lim)
posted by Tommy feels dumb on .
consider the statement
lim x³6x²+11x6 / x1 = 2
x>1
Using the definition of the limit, state what must be true for the above limit to hold, that is, for
every ..., there is ..., so that.... Use a specific function and limit not just f and L.
Verify the limit is true by finding ä as an expression of ϵ.
Draw a picture illustrating the relation between ϵ, ä and the function.
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x^36x^2+11x6 = (x1)(x2)(x3)
So, for all x≠1,
f(x) = (x2)(x3)
as x>1, f(x)>2 since both factors are negative
we need to show that for every ϵ>0 there is a δ such that
f(x+δ)2 < ϵ
we can dispense with the absolute value stuff, since f(x) > 0 and we are taking the upper limit. So, we just need to show that we can solve for δ, no matter which small ϵ we choose.
((x+δ)2)((x+δ)3)2 < ϵ
That's just a simple quadratic, which will have two real roots.