Problem 32 - 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. $$\lim _{h \rightarrow 0} \frac{100}{(10 h-1)^{11}+2}$$

Step-by-Step Solution

Answer

Answer: The limit is 100.

Step 1: Substitute h with 0

Since we want to determine the value for which our expression approaches as \(h\) goes to \(0\), we can directly substitute \(h=0\) into the expression. $$\lim _{h \rightarrow 0} \frac{100}{(10h-1)^{11}+2} = \frac{100}{(10(0)-1)^{11}+2}$$

Step 2: Simplify the Expression

Now, we will simplify the expression by evaluating the terms inside the brackets, and the exponent. $$\frac{100}{(10(0)-1)^{11}+2} = \frac{100}{(-1)^{11}+2}$$

Step 3: Evaluate the Limit

Since \((-1)^{11} = -1\), our simplified expression becomes: $$\frac{100}{(-1) + 2} = \frac{100}{1} = 100$$ Therefore, the limit of the given expression as \(h\) approaches \(0\) is: $$\lim _{h \rightarrow 0} \frac{100}{(10 h-1)^{11}+2} = 100$$

Key Concepts

Limit Evaluation

Calculus often involves determining how a function behaves as the input approaches a specific value. This process is known as limit evaluation. Imagine observing a car approaching a stop sign: you want to predict how it stops. Similarly, limit evaluation tells us what value a function approaches.

To evaluate limits, follow these principles:

• Direct Substitution: If substituting the approaching value into the function doesn't result in an undefined form like \( \frac{0}{0} \), the substitution provides the limit.
• Simplification: If direct substitution fails due to an undefined form, simplify the function. Techniques such as factoring, expanding, or cancelling common terms can help.
• Infinite Limits: Recognize when a limit approaches infinity, which suggests that a function grows very large or very small.

Understanding these approaches and applying them systematically helps in finding limits effectively. In our example, by substituting \( h = 0 \) directly, the result is a straightforward calculation indicating the limit of \( 100 \).

Substitution in Limits

Substitution is often the first method employed in finding limits. It's as simple as placing the value you are approaching into the function, much like putting a key into a lock to see if it fits.

Here's how substitution is used in limits:

• Ensure Continuity: When a function is continuous at the point you are investigating, direct substitution will work nicely. Continuity means there are no jumps or holes at that point.
• Smooth Solutions: By substituting \( h \) with 0 as in our problem, you can directly evaluate the expression. If this computation results in a real number, you've found your limit.
• Exceptional Cases: Some cases such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) are indeterminate and require further methods to resolve.

Thus, substitution is a handy tool, and in our sample problem, simply placing \( h = 0 \) into the equation delivers the limit immediately.

Real Numbers

Real numbers are the values that encompass all the numbers students encounter on a daily basis, both rational and irrational. In the context of limits, real numbers allow us to evaluate the behavior of functions at specific points.

Key properties include:

• Rational Numbers: These include fractions and integers. They have clear and defined limits unless approaching an undefined point.
• Irrational Numbers: Numbers like \( \sqrt{2} \) or \( \pi \) can't be expressed as simple fractions, yet they still operate within limits.
• Application in Limits: Limits can frequently result in real numbers, suggesting the function settles to this real value at a given input.

In the exercise provided, knowing that \( a, b, c, \) and \( k \) are fixed real numbers helps assure that any evaluation stays within the confines of real-number arithmetic, leading to clean and interpretable results like the limit of 100.