Find the limit.
lim x-->1 (x^3)-1/((5x^2)+2x-7)
Please show steps.
= lim (x-1)(x^2 + x + 1)/((x-1)(5x+7)) as x ---> 1
= lime (x^2 + x + 1)/(5x+7) , as x --> 1
= 3/12
= 1/4
Thanks.
To find the limit of the given function as x approaches 1, we can use direct substitution. However, if substituting 1 results in an undefined expression (such as dividing by zero), we need to employ a different approach.
1. Start by substituting the value of x into the function:
lim x->1 ((x^3)-1)/((5x^2)+2x-7)
2. Since substituting x = 1 does not yield an undefined expression, we can directly evaluate the function at x = 1:
((1^3)-1)/((5(1^2))+2(1)-7)
(1-1)/(5+2-7)
0/0
At this point, the expression becomes indeterminate as it yields a fraction of 0/0. To resolve this, we need to factorize the numerator and denominator and simplify further.
3. Factorize the numerator using the difference of cubes formula:
(x^3 - 1) = (x - 1)(x^2 + x + 1)
4. Factorize the denominator:
(5x^2 + 2x - 7) = (x + 1)(5x - 7)
5. Simplify the expression further:
((x - 1)(x^2 + x + 1))/((x + 1)(5x - 7))
6. Since we canceled a common factor between the numerator and denominator, we can now try substituting x = 1 again:
((1 - 1)(1^2 + 1 + 1))/((1 + 1)(5(1) - 7))
(0)(3)/(2)(-2)
0/(-4)
The expression simplifies to 0 divided by -4, which equals 0.
Therefore, the limit as x approaches 1 is 0.