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Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)
Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)
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Practice Questions
1 question
Q1
Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)
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3
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Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). Canceling (x - 1) gives lim (x -> 1) (x^2 + x + 1) = 3.
Questions & Step-by-step Solutions
1 item
Q
Q: Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)
Solution:
Factoring gives (x - 1)(x^2 + x + 1)/(x - 1). Canceling (x - 1) gives lim (x -> 1) (x^2 + x + 1) = 3.
Steps: 7
Show Steps
Step 1: Identify the limit you need to calculate: lim (x -> 1) (x^3 - 1)/(x - 1).
Step 2: Notice that if you plug in x = 1 directly, both the numerator and denominator become 0. This means we need to simplify the expression.
Step 3: Factor the numerator (x^3 - 1). It can be factored as (x - 1)(x^2 + x + 1).
Step 4: Rewrite the limit using the factored form: lim (x -> 1) [(x - 1)(x^2 + x + 1)/(x - 1)].
Step 5: Cancel the (x - 1) in the numerator and denominator. This gives you lim (x -> 1) (x^2 + x + 1).
Step 6: Now, plug in x = 1 into the simplified expression: 1^2 + 1 + 1 = 3.
Step 7: Therefore, the limit is 3.
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