Limite de una función real de variable real

Summary

In this video, the process of calculating the limit of a rational function as x approaches 2 is explained. The function involves a polynomial quotient, and initially, direct substitution results in a denominator of zero. The script explores how approaching the limit leads to infinity, as dividing by a very small number yields a very large result. The sign of the infinity (positive or negative) depends on whether x approaches 2 from the left or right. Though the video doesn't focus on left-hand or right-hand limits, it suggests that in other videos, this analysis will be explored.

The video is abnormal, and we are working hard to fix it.
Please replace the link and try again.

Q & A

What is the limit being calculated in this transcript?

The limit being calculated is as x approaches 2 for the expression (x + 3) / (x - 2).

Why do we initially substitute x = 2 into the expression?

We substitute x = 2 into the expression because the function is a ratio of polynomials, which are continuous functions. If substituting yields a valid result, that value would be the limit.

What happens when we substitute x = 2 into the expression?

When we substitute x = 2, the numerator becomes 2 + 3 = 5, and the denominator becomes 2 - 2 = 0, leading to an indeterminate form (5/0).

Why does a denominator of 0 cause an issue?

A denominator of 0 is undefined in mathematics because division by zero does not result in a valid number. This suggests we need to analyze the behavior of the function more carefully.

What does 'x approaching 2' mean in the context of a limit?

'x approaching 2' means that x gets infinitely close to 2 but does not actually reach 2. The limit describes the behavior of the function as x gets closer to 2.

How do we interpret the expression when x is close to 2 but not exactly 2?

When x is close to 2, the denominator becomes a very small number approaching zero, but not zero itself. This leads to a very large or infinite result, depending on whether x approaches 2 from the left or the right.

What does dividing by a number close to zero imply for the result?

Dividing by a number very close to zero, whether positive or negative, results in a very large value. If the denominator is a small positive number, the result tends toward positive infinity; if negative, it tends toward negative infinity.

What role does the sign of the denominator play in determining the limit?

The sign of the denominator affects whether the result tends toward positive or negative infinity. A small positive denominator leads to positive infinity, while a small negative denominator leads to negative infinity.

What does the phrase 'approaching from the left' or 'approaching from the right' mean?

Approaching from the left means that x is getting closer to 2 from values smaller than 2 (from the negative side), while approaching from the right means that x is approaching 2 from values greater than 2 (from the positive side).

Will the limit always result in infinity for this case?

Yes, the limit will always result in infinity, but whether it's positive or negative infinity depends on whether x approaches 2 from the left or the right. The exact behavior depends on the direction of approach.