evaluate limx->c (x^4-c^4)/(x^2-c^3)
what rubbish!
you cannot factor out x^2-c^2 from the bottom!
substituting in x=c directly,
limx->c (x^4-c^4)/(x^2-c^3) = (c^4-c^4)/(c^2-c^3) = 0
because the top has a higher degree than the bottom.
My apologies for the incorrect factorization in my previous response. You are correct in pointing out that x^2 - c^2 cannot be factored out from the denominator.
Using direct substitution by substituting x = c:
lim(x->c) (x^4 - c^4)/(x^2 - c^3) = (c^4 - c^4)/(c^2 - c^3) = 0
The numerator becomes 0 and the denominator approaches 0 as c approaches a certain value. Therefore, the limit of (x^4 - c^4)/(x^2 - c^3) as x approaches c is 0.
To evaluate this limit, we can directly substitute the value of c into the expression and see what we get.
Let's substitute x = c:
lim(x->c) (x^4 - c^4)/(x^2 - c^3) = (c^4 - c^4)/(c^2 - c^3)
Notice that the numerator becomes 0, while the denominator approaches 0 as c approaches a certain value. This indicates a potential issue with direct substitution.
To further evaluate this limit, we can factor out a common factor from both the numerator and denominator.
(x^4 - c^4) = (x^2 + c^2)(x^2 - c^2)
(x^2 - c^3) cannot be factored any further.
Substituting these into the expression again:
lim(x->c) (x^4 - c^4)/(x^2 - c^3) = lim(x->c) [(x^2 + c^2)(x^2 - c^2)]/(x^2 - c^3)
Now, notice that (x^2 - c^2) terms can cancel out, leaving the expression as:
lim(x->c) [(x^2 + c^2)] = c^2 + c^2 = 2c^2
Therefore, the limit of (x^4 - c^4)/(x^2 - c^3) as x approaches c is 2c^2.