Question

If function \(f:(1,3) \rightarrow R\) satisfies the inequality \(\frac{\cos (2 x-6)-37}{2}

Short answer

The limit cannot be determined from the given inequality.

Step-by-step solution

Determine the left-hand side of the inequality

First, analyze the expression on the left-hand side of the inequality: \ \(\frac{\cos(2x-6)-37}{2}\). \ Evaluate the behavior of this function as \(x\) approaches 3 from the left. When \(x = 3\), \ \(\cos(2 \cdot 3 - 6) = \cos 0 = 1\), so \(\frac{1 - 37}{2} = \frac{-36}{2} = -18\). Therefore, as \(x\) approaches 3 from the left, the left-hand side approaches -18.

Determine the right-hand side of the inequality

Next, consider the expression on the right-hand side of the inequality: \ \(\frac{x^{2} |2x-6|}{x-3}\). \ Simplify this expression for \(x\) close to 3. Notice that \(|2x-6| = 2|x-3|\). Thus, the expression can be written as: \ \(\frac{x^{2} \cdot 2|x-3|}{x-3} = 2x^{2}\). As \(x\) approaches 3 from the left, \(2x^{2}\) approaches \(2(3^{2}) = 18\). Therefore, as \(x\) approaches 3 from the left, the right-hand side approaches 18.

Analyze the limits of the function

Given the inequality: \ \(\frac{\cos(2x-6)-37}{2} < f(x) < \frac{x^{2} |2x-6|}{x-3}\), \ we analyzed both sides as \(x\) approaches 3 from the left. The inequality becomes: \ \(-18 < f(x) < 18\) as \(x\) approaches 3 from the left. \ Therefore, the limit of \(f(x)\) as \(x\) approaches 3 from the left is constrained by these bounds.

Conclude the limit behavior of the function

Analyze the limit within the bounds: \ Since \( -18 < f(x) < 18\) as \(x\) approaches 3 from the left, \(f(x)\) can take any value within this interval. Hence, \(\lim_{{x \to 3^{-}}} f(x)\) cannot be determined exactly from the given inequality because the exact value within these bounds is not specified.

Key concepts

inequalities in functions

Inequalities in functions are useful to understand how a function behaves within a specific range. Consider the inequality provided in the exercise: \(\frac{\text{cos}(2x-6)-37}{2} < f(x) < \frac{x^{2}|2x-6|}{x-3}\text{ for all }x \text{ in }(1,3)\). \ This inequality tells us the bounds within which the function \(f(x)\) falls for x in the interval \( (1,3)\).

To make this clearer:

• The term on the left side \(\frac{\text{cos}(2x-6)-37}{2}\) represents the lower bound of the function. It establishes the minimum value that \(f(x)\) can attain within the given range.
• The term on the right side \(\frac{x^{2}|2x-6|}{x-3}\) represents the upper bound. It establishes the maximum value that \ f(x) \ can attain within the same range.

As \(x\) approaches different values in the interval \( (1,3)\), the values of these expressions can change, establishing different possible ranges for \ f(x) \. By analyzing these bounds, we can get valuable insights into the behavior of the function within the given domain.

limit behavior

Understanding the limit behavior of a function is crucial in calculus. Limits describe the value that a function approaches as the input approaches a certain point. In the context of this exercise, we are interested in the behavior of the function \( f(x) \) as \( x \) approaches 3 from the left.

To analyze this:

• First, we need to evaluate the behavior of the left-hand side expression \(\frac{\text{cos}(2x-6)-37}{2}\). When \ x \ approaches 3 from the left, the value of this expression tends to -18.
• Next, we consider the right-hand side expression \ \frac{x^{2}|2x-6|}{x-3} \. When \ x \ approaches 3 from the left, the value of this expression tends to 18.

Given these observations, the inequality simplifies to:

\(-18 < f(x) < 18\) as \ x \ approaches 3 from the left. This range tells us that \ f(x) \ is bounded between -18 and 18, but the exact limit cannot be pinpointed without more information. Therefore, from the inequality alone, we can conclude that the limit of \ f(x) \ as \ x \ approaches 3 from the left is undetermined.

continuity analysis

Continuity analysis helps us understand how a function behaves at specific points and intervals. A function is said to be continuous at a point if there is no break, jump, or hole at that point. To analyze continuity, we often use limits.

In the given exercise, we are dealing with the interval \(1,3\) and evaluating the behavior of \( f(x) \) as \ x \ approaches 3 from the left.

To determine continuity:

• Evaluate the left and right-hand expression limits as \ x \ approaches the point in question (in this case, 3).
• If the limits converge to the same value, \ f(x) \ is continuous at that point.

In our exercise, the limits of the input expression range from \-18\ to \18\ as \ x \ approaches 3 from the left. However, we can't pinpoint an exact limit due to the undefined gap between -18 and 18. This suggests that we can't guarantee continuity solely based on the provided inequality. Hence, additional information about \( f(x) \) is needed to conclude its continuity at \( x = 3 \).