The value of the given limit lim ₓ → 1 (x³ - 1)/(x - 1) = <u>3</u>.
In the question, we are asked what is the limit of startfraction x cubed minus 1 over x minus 1 endfraction as x approaches 1, that is, lim ₓ → 1 (x³ - 1)/(x - 1).
Unique real numbers are limits in mathematics. Let us consider a real-valued function "f" and a real number "c", the limit is normally defined as lim ₓ → c f(x) = L. The phrase is to be understood as "the limit of f of x, as x approaches c equals L." The right arrow indicates that function f(x) approaches the limit L as x approaches c. The "lim" displays the limit.
The value of the given limit can be calculated as follows:
lim ₓ → 1 (x³ - 1)/(x - 1)
= lim ₓ → 1 (x³ - 1³)/(x - 1), Since 1³ = 1
= lim ₓ → 1 (x - 1)(x² + x + 1)/(x - 1) , using the formula: a³ - b³ = (a - b)(a² + ab + b²)
= lim ₓ → 1 (x² + x + 1), Cancelling (x - 1) in the numerator and the denominator, since x → 1, that is, x is approaching 1 and not equal to 1, implying that x - 1 ≠ 0.
= (1² + 1 + 1), Substituting x = 1, and removing the limit.
= 3.
Thus, the value of the given limit lim ₓ → 1 (x³ - 1)/(x - 1) = <u>3</u>.
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