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Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)
Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)
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Practice Questions
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Q1
Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)
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Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). Canceling gives lim (x -> 1) (x^3 + x^2 + x + 1) = 4.
Questions & Step-by-step Solutions
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Q
Q: Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)
Solution:
Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). Canceling gives lim (x -> 1) (x^3 + x^2 + x + 1) = 4.
Steps: 9
Show Steps
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.
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