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Evaluate the limit lim x->1 of (x^3 - 1)/(x - 1).
Evaluate the limit lim x->1 of (x^3 - 1)/(x - 1).
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Practice Questions
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Q1
Evaluate the limit lim x->1 of (x^3 - 1)/(x - 1).
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Factoring gives (x-1)(x^2 + x + 1)/(x-1) = x^2 + x + 1, thus limit is 3.
Questions & Step-by-step Solutions
1 item
Q
Q: Evaluate the limit lim x->1 of (x^3 - 1)/(x - 1).
Solution:
Factoring gives (x-1)(x^2 + x + 1)/(x-1) = x^2 + x + 1, thus limit is 3.
Steps: 8
Show Steps
Step 1: Identify the limit we want to evaluate: lim x->1 of (x^3 - 1)/(x - 1).
Step 2: Notice that if we plug in x = 1 directly, both the numerator and denominator become 0, which is an indeterminate form.
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 of [(x - 1)(x^2 + x + 1)]/(x - 1).
Step 5: Cancel the (x - 1) terms in the numerator and denominator, as long as x is not equal to 1.
Step 6: Now we have lim x->1 of (x^2 + x + 1).
Step 7: Substitute x = 1 into the simplified expression: 1^2 + 1 + 1 = 3.
Step 8: Therefore, the limit is 3.
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