Understanding One-Sided Limits and Cosine Function

The video tutorial explores the function f(x) = x / (1 - cos(x - 2)) and its behavior at x = 2. The function is undefined at this point, prompting an investigation into the one-sided limits as x approaches 2. The tutorial discusses two methods: analyzing the function's properties without a calculator and using a calculator to create a table of values. The analysis reveals that as x approaches 2 from either side, the function's value becomes unbounded, heading towards positive infinity. The tutorial concludes by confirming this behavior through logical deduction and mathematical reasoning.

What is the main issue with evaluating f(x) at x=2?

• The function is undefined at x=2.

• The function is zero at x=2.

• The function is negative at x=2.

• The function is infinite at x=2.

Which method is suggested for finding the one-sided limits of f(x) without a calculator?

• Inspecting the properties of the cosine function

• Using a graphing tool

• Applying the quadratic formula

• Using a logarithmic table

What is the behavior of the cosine function as x approaches 2 from the positive direction?

• It becomes greater than one.

• It becomes negative.

• It remains constant.

• It approaches one but stays less than one.

As x approaches 2 from the positive side, what happens to the value of f(x)?

• It becomes negative.

• It approaches zero.

• It remains constant.

• It becomes unbounded in the positive direction.

What is the significance of the numerator being positive as x approaches 2?

• It indicates the function will be negative.

• It suggests the function will be zero.

• It implies the function will be positive.

• It has no effect on the function's behavior.

How does the function behave as x approaches 2 from the negative direction?

• It approaches zero.

• It approaches negative infinity.

• It becomes unbounded in the positive direction.

• It remains constant.

What is the relationship between cosine of negative values and positive values?

• Negative values are always greater.

• Positive values are always greater.

• They are always different.

• They are equal.

What conclusion is drawn about the function's behavior as x approaches 2 from both sides?

• The function remains constant.

• The function approaches zero.

• The function becomes unbounded in the negative direction.

• The function becomes unbounded in the positive direction.

Why does the denominator become smaller as x approaches 2?

• Because cosine approaches zero.

• Because the numerator increases.

• Because cosine approaches one.

• Because the numerator decreases.

What happens to the value of f(x) as the denominator gets smaller?

• It becomes unbounded towards infinity.

• It becomes negative.

• It remains constant.

• It approaches zero.