Problem 61 - Calculus Early Transcendentals

Question

Evaluate the following limits, where \(c\) and \(k\) are constants. \(\lim _{x \rightarrow 2}(5 x-6)^{3 / 2}\)

Step-by-Step Solution

Answer

Answer: The value of the limit as x approaches 2 for the expression (5x - 6)^(3/2) is 8.

Step 1: Try Direct Substitution

Check if the given function should be directly substituted by replacing \(x\) with \(2\): \((5(2)-6)^{3 / 2} = (4)^{3 / 2}\). Since this expression is well-defined and produces a finite value, we can evaluate the limit directly using substitution.

Step 2: Evaluate Limit Using Direct Substitution

Now, replace \(x\) with \(2\) in the expression to evaluate the limit: \(\lim _{x \rightarrow 2}(5 x-6)^{3 / 2} = (5(2)-6)^{3 / 2} = (4)^{3 / 2} = 2^{3} = 8\). The limit of the expression as \(x\) approaches 2 is 8.

Key Concepts

Direct Substitution

When faced with a limit problem, one of the simplest methods you can use is direct substitution. This involves replacing the variable in the function with the value it approaches. It's a quick check to see if the limit can be easily determined without further algebraic manipulation.
For direct substitution to work:

• The function must be defined at the point you're substituting. If it's not, direct substitution can't help and you'll need another method.
• The expression should not result in an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

In our example, the function \((5x-6)^{3/2}\) allows for a direct substitution since plugging \(x = 2\) does not create any undefined behavior. It yields a clear numeric result.

Continuous Functions

A continuous function is a function that is smooth and unbroken; graphically, you can draw it without lifting your pen from the paper. Continuity is key when evaluating limits, especially when using direct substitution.
Mathematically, a function \(f(x)\) is continuous at a point \(x = c\) if:

• \(f(c)\) is defined.
• The limit of \(f(x)\) as \(x\) approaches \(c\) exists.
• The limit of \(f(x)\) as \(x\) approaches \(c\) equals \(f(c)\).

In simpler terms, there should be no jumps, holes, or vertical asymptotes at that point. This notion of continuity allows us to confidently use direct substitution to evaluate limits, as with the given function \((5x-6)^{3/2}\), which is continuous around the point \(x = 2\).

Evaluating Limits

Evaluating limits is a fundamental concept in calculus that helps us understand the behavior of functions as they approach specific points. The process usually involves finding out what value (if any) your function gets closer to as the input approaches a certain point.
There are various strategies to evaluate limits:

• **Direct Substitution**: As mentioned, the easiest method when applicable. Use it if the function is continuous and not resulting in indeterminate forms.
• **Factorization**: Break down complex expressions to simplify and remove indeterminate forms.
• **Rationalizing**: Particularly useful for limits involving square roots or radical expressions.
• **L'Hôpital's Rule**: When direct substitution leads to \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), this rule can be applied.

For our example, since the function is continuous and does not lead to an indeterminate form when \(x = 2\) is substituted, evaluating the limit was straightforward with direct substitution.