Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)

Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)

Step 1: Identify the limit we need to calculate: lim (x -> 1) (x^4 - 1)/(x - 1).
Step 2: Notice that both the numerator (x^4 - 1) and the denominator (x - 1) become 0 when x = 1. This means we can factor the numerator.
Step 3: Factor the numerator x^4 - 1. It can be factored as (x^2 - 1)(x^2 + 1).
Step 4: Further factor (x^2 - 1) into (x - 1)(x + 1). So, we have: (x - 1)(x + 1)(x^2 + 1).
Step 5: Now rewrite the limit: lim (x -> 1) [(x - 1)(x + 1)(x^2 + 1)]/(x - 1).
Step 6: Cancel the (x - 1) in the numerator and denominator. This gives us: lim (x -> 1) (x + 1)(x^2 + 1).
Step 7: Now substitute x = 1 into the remaining expression: (1 + 1)(1^2 + 1).
Step 8: Calculate (1 + 1) = 2 and (1^2 + 1) = 2. So, we have: 2 * 2 = 4.
Step 9: Therefore, the limit is 4.

Limit Calculation – Understanding how to evaluate limits, especially when faced with indeterminate forms.
Factoring Polynomials – The ability to factor polynomials to simplify expressions before taking limits.
Cancellation of Terms – Recognizing when and how to cancel common factors in a limit expression.