Límites por racionalización 1

Summary

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Q & A

What is the goal of the problem discussed in the script?

The goal of the problem is to calculate the limit of the function 3z - 6 over 1 - √(4z - 7) as z tends to 2 using the technique of rationalization.

Why is the initial evaluation of the limit indeterminate?

The initial evaluation results in the indeterminate form 0/0. This happens because substituting z = 2 into both the numerator and the denominator results in 0, leading to the indeterminate form.

What technique is suggested to resolve the indeterminate form?

The technique of rationalization is suggested to resolve the indeterminate form. This involves multiplying the expression by the conjugate of the denominator.

What is the conjugate of the denominator in this case?

The conjugate of the denominator, 1 - √(4z - 7), is 1 + √(4z - 7). The technique involves multiplying both the numerator and the denominator by this conjugate.

How does multiplying by the conjugate simplify the denominator?

Multiplying by the conjugate converts the denominator into a difference of squares, which simplifies to 1² - (√(4z - 7))² = 1 - (4z - 7).

What expression is obtained after applying the difference of squares formula?

After applying the difference of squares formula, the denominator becomes 1 - (4z - 7), which simplifies to -4z + 8.

How is the indeterminate form further simplified?

The indeterminate form is further simplified by factoring out common terms in both the numerator and denominator. In this case, a factor of 3 is taken from the numerator, and a factor of -4 is taken from the denominator.

What happens to the factors after simplification?

After factoring, the common term (z - 2) appears in both the numerator and denominator, allowing it to be canceled out. This resolves the indeterminate form.

What is the final step in evaluating the limit?

The final step is to substitute z = 2 into the simplified expression, which results in 3 * (1 + 1) / -4, simplifying to -3/2.

What is the final result of the limit as z approaches 2?

The final result of the limit as z approaches 2 is -3/2.