What is the
lim(x-3/x^3-27)
x->3
the denominator factors as a difference of cubes
so lim (x-3)/(x^3 - 27) as x---> 3
= lim (x-3)/[(x-3)(x^2 + 3x + 9)] as x ---> 3
= lim 1/((x^2 + 3x + 9)
= 1/(9+9+9)
= 1/27
To find the limit of a function as x approaches a specific value, in this case x=3, we can use direct substitution to evaluate the function at that value. However, this particular function is undefined at x=3 because it results in 0/0, which is an indeterminate form.
To determine the limit in such cases, we can try factoring the expression and simplifying it. The numerator in this case, x-3, can be factored using the difference of cubes formula: a^3 - b^3 = (a - b)(a^2 + ab + b^2). Thus, x-3 becomes (x-3)(x^2 + 3x + 9).
Simplifying further, we have [(x-3)(x^2 + 3x + 9)] / (x^3 - 27).
Now, we can cancel out the common factor of (x-3) in the numerator and the denominator:
[(x-3)(x^2 + 3x + 9)] / [(x-3)(x^2 + 3x + 9)]
Notice that (x^2 + 3x + 9) is still present in the numerator and denominator. At this point, we no longer have a 0/0 indeterminate form, and we can further simplify by canceling out these factors:
1
Therefore, the limit of the function lim(x-3/x^3-27) as x approaches 3 is 1.