Problem 31 - Calculus: Early Transcendentals

Question

Find the following limits or state that they do not exist. Assume \(a, b, c,\) and k are fixed real numbers. $$\lim _{x \rightarrow 2}(5 x-6)^{3 / 2}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit as x approaches 2 of \((5x-6)^{3/2}\). Answer: The limit as x approaches 2 of \((5x-6)^{3/2}\) is 8.

Step 1: Direct Substitution

To find the limit as \(x \rightarrow 2\), we can directly substitute \(x=2\) into the expression \((5x-6)^{3/2}\): $$ (5(2)-6)^{3/2} $$

Step 2: Simplify Expression

Now we just simplifying the expression: $$ (10-6)^{3/2} $$ $$ (4)^{3/2} $$

Step 3: Evaluate

Now we can calculate the expression: $$ (2^2)^{3/2} = 2^{2 \cdot 3 / 2} = 2^3 $$ $$ 2^3 = 8 $$ So the limit is 8: $$ \lim _{x \rightarrow 2} (5x-6)^{3 / 2} = 8 $$

Key Concepts

Direct Substitution

When evaluating limits in calculus, one of the simplest and most straightforward methods is known as direct substitution. It is exactly as it sounds – you substitute the value that x approaches directly into the function, if the function is continuous at that point. This tactic is quick and effective when the function doesn’t lead to an indeterminate form like 0/0 or \/infty/\infty.

For the given problem, as X approaches 2, we can substitute 2 into the expression \(5x-6)^{3/2}\) without any complications. Assuming no division by zero or subtraction of infinities occurs, direct substitution provides the limit promptly. This approach relies on an understanding that a continuous function's graph has no breaks, holes, or jumps at the point of interest – resulting in a reliable limit value.

Limit Laws

In the process of evaluating limits, we often utilize fundamental rules known as limit laws. These essential tools allow us to break down complex expressions into simpler components, which can then be handled separately. Some of these laws include the sum law, product law, quotient law, and power law.

Applying these laws systematically can simplify the process of finding limits, especially when directly substituting the approaching value does not immediately yield an answer. Moreover, limit laws preserve the algebraic structure of the functions involved and thus, they accommodate the simplification needed for direct substitution. These laws affirm that the operation performed after taking limits yields the same result as performing the operation first and then taking the limit, provided the limits exist and are finite.

Simplifying Expressions

To effortlessly navigate through algebraic expressions when determining limits, one needs to master the art of simplifying expressions. The process often involves rearranging and reconfiguring algebraic terms into their most manageable form, typically through operations such as factoring, canceling, expanding, and combining like terms.

In our exercise, after direct substitution, simplification is key to evaluate \(4)^{3/2}\) effectively. Breaking the expression down and rewriting it in simpler terms, or in some instances, as an equivalent expression without radicals, facilitates an easier path to the solution. The idea is to transform the problem into a format where the limit can be readily evaluated with minimal algebraic manipulation.

Radical Exponents

Understanding radical exponents is crucial when working with both limits and algebraic simplification. Radical expressions often appear in functions involving roots, such as square roots, cube roots, etc. The exponentiation of radicals can be interpreted through the lens of powers and their properties.

For instance, the expression \(a^b)^c\) is equivalent to \(a^{b*c})\), which can simplify computations. In the exercise, the expression \(2^2)^{3/2}\) can be simplified using exponent rules to \(2^3)\), which then allows for easy evaluation. It’s important to become familiar with these rules, as they can significantly streamline the process of calculating limits that involve radical exponents.