Problem 32 - Calculus: Early Transcendentals

Question

Evaluate limit and justify your answer. $$\lim _{x \rightarrow 2}\left(\frac{3}{2 x^{5}-4 x^{2}-50}\right)^{4}$$

Step-by-Step Solution

Answer

**Question:** Evaluate the following limit and justify your answer: $$\lim _{x \rightarrow 2}\left(\frac{3}{2x^5 - 4x^2 - 50}\right)^4$$ **Answer:** The limit is: $$\lim _{x \rightarrow 2}\left(\frac{3}{2 x^{5}-4 x^{2}-50}\right)^{4} = \frac{81}{16}$$ We justified our answer by substituting the value of x into the given function, simplifying the expression, and evaluating the final result.

Step 1: Substitute the value of x into the given function

First, substitute the value of x (which is 2) into the given function: $$\left(\frac{3}{2(2)^5 -4(2)^2 - 50}\right)^4$$

Step 2: Simplify the expression

Now, we simplify the expression: $$\left(\frac{3}{2(32) - 4(4) - 50}\right)^4$$ $$\left(\frac{3}{64 - 16 - 50}\right)^4$$ $$\left(\frac{3}{-2}\right)^4$$

Step 3: Evaluate the limit and justify the answer

Finally, evaluate the limit and justify the result: $$\left(\frac{3}{-2}\right)^4 = \left(-\frac{3}{2}\right)^4 = \frac{81}{16}$$ So, the limit is: $$\lim _{x \rightarrow 2}\left(\frac{3}{2 x^{5}-4 x^{2}-50}\right)^{4} = \frac{81}{16}$$ We have justified our answer by substituting the value of x into the given function, simplifying the expression, and evaluating the final result.

Key Concepts

Limit of a Function

When studying calculus, one of the fundamental concepts you'll encounter is the limit of a function. A limit tries to find the value that a function approaches as the input (or 'x' value) gets closer to a certain number. It's a way of asking, 'What value does this function aim to reach as x moves toward a specific point?'

Understanding the concept of limits is crucial because they allow you to handle situations where the function doesn't actually reach that value or where it's not clearly defined at that point - for example, in cases of discontinuity or infinity. In the given exercise, the limit is being evaluated as x approaches the value of 2.

In evaluating the limit of the function, the initial step is typically to plug in the approaching value into the function, if possible. This often gives a clear understanding of the function's behavior near the point of interest. However, if this direct substitution results in an undefined expression (like a division by zero or an indeterminate form), other techniques such as factoring, expanding, or trigonometric identities might be used to simplify the expression and find the limit.

Calculus Problems

Solving calculus problems often requires a combination of insight, knowledge, and a systematic approach. When faced with a problem like evaluating a limit, it's important not to rush but to consider the variety of tools and rules at your disposal.

One common technique you'll use in calculus is simplification, which allows you to break down complex expressions into more manageable pieces. It's also essential to understand and recognize when to apply specific rules such as the L'Hopital's rule when figuring out indeterminate forms or recognizing when a limit does not exist.

Practical Tips for Tackling Calculus Problems:

• Read the problem carefully to understand what is being asked.
• Identify which calculus concepts apply to the problem at hand.
• Simplify expressions whenever possible to make the calculations more straightforward.
• Check your work as you go to avoid errors.
• Practice regularly to recognize common patterns and solutions.

With regards to the problem presented, by carefully substituting and simplifying, we can find the exact behavior of the function as x approaches 2, leading us to the correct limit.

Simplifying Expressions

An indispensable skill in calculus and algebra is simplifying expressions. This process involves reducing an expression to its simplest form, making it easier to understand or further manipulate mathematically. This could involve combining like terms, using algebraic identities, factoring, expanding, or canceling out common terms in fractions.

To simplify effectively, it's critical to have a firm grasp of basic algebraic principles and operations. Always pay attention to the order of operations (remember PEMDAS: parentheses, exponents, multiplication and division, addition and subtraction) to avoid common mistakes.

Why Simplify Expressions?

Simplification can reveal properties of the expression that aren't immediately apparent and can make it more visually apparent what the behavior of the function is as x approaches a given value. In the context of our exercise, simplification was key to turning an intimidating fraction into a straightforward calculation and thus reaching the final answer without ambiguity.