Problem 22 - Calculus: Early Transcendentals

Question

Guess the value of the limit (if it exists) by evaluating the function at the given numbers (correct to six decimal places). \( \displaystyle \lim_{h \to 0}\frac{(2+h)^5 - 32}{h} \), \( h = \pm 0.5, \pm 0.1, \pm 0.01, \pm 0.001, \pm 0.0001 \)

Step-by-Step Solution

Answer

The limit is 80.

Step 1: Substitute h-values into the function

First, substitute the different values of \( h \) into the expression \( \frac{(2+h)^5 - 32}{h} \). Begin by substituting the largest values (\( h = \pm 0.5 \)) and gradually reduce \( h \) to the smallest values (\( h = \pm 0.0001 \)). Each calculation will approximate the limit as \( h \) approaches zero.

Step 2: Simplify the expression for each substitution

For each \( h \) value, calculate \( (2+h)^5 \) to simplify the numerator of the function. For example:- When \( h = 0.5 \), calculate \((2+0.5)^5 - 32\);- Similarly, for \(h = -0.5\), \((2-0.5)^5 - 32\), and so on.This calculation should be done for all given \( h \) values to prepare for the division.

Step 3: Perform division for each expression

Now divide the simplified numerator from Step 2 by \( h \) for each case:- For \( h = 0.5 \), divide by 0.5;- For \( h = -0.5 \), divide by -0.5;- Continue similarly for all other values.Ensure each result is correct to six decimal places.

Step 4: Analyze the results

After calculating the expression for each \( h \), observe the trends in the results as \( h \) approaches zero. For positive and negative \( h \) close to zero, the results should increasingly stabilize around a certain number. This number will approximate the limit.

Step 5: Determine the limit

Conclude by stating that the value the function approaches as \( h \) gets very close to zero, which is observed from the results across various \( h \) values. This value is the answer to the limit \( \displaystyle \lim_{h \to 0}\frac{(2+h)^5 - 32}{h} \).

Key Concepts

Numerical Approximation

When faced with finding limits, sometimes exact calculation isn't possible at first. This is where numerical approximation comes in handy. For students, it's about evaluating how a function behaves as it approaches a specific point — in this case, as \( h \) approaches zero in the given limit \( \displaystyle \lim_{h \to 0}\frac{(2+h)^5 - 32}{h} \).
By plugging in values of \( h \) that get progressively closer to zero, you observe how the function's output stabilizes. This helps you make an educated guess about the limit value.
Some tips for numerical approximation:

• Start with larger increments, then use smaller ones for precision.
• Show consistency across both negative and positive values.
• Ensure calculations are accurate; even small errors can lead to incorrect conclusions.

Ultimately, numerical approximation offers a practical method to estimate limits without initially needing algebraic manipulation.

Derivative Calculation

Limits are fundamental to understanding derivatives, which measure how a function changes at any given point. The exercise actually hides a derivative in disguise! It's tied to the basic definition of a derivative: \( \displaystyle f'(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h} \).
This mirrors our task, since \( x = 2 \) and \( f(x) = x^5 \), making the expression \( \displaystyle \lim_{h \to 0}\frac{(2+h)^5 - 32}{h} \) a derivative problem.
For effective derivative calculation, remember:

• This limit gives the slope or rate of change of \( f \) at \( x \).
• Understanding the derivative conceptually aids in better grasping any given problem.
• The function's behavior near the point of tangency is revealed through such calculations.

It ties together understanding calculus' real-world problems with algebraic functions.

Limits and Continuity

Limits serve as a foundation for studying continuity in calculus. A function is continuous at a point \( a \) if the limit of the function as it approaches \( a \) is the same as its value at \( a \). But what if we're focused on the precisely the point it approaches?
When you perform exercises like the one provided, it enlightens you about the game's rule: as a function nears a specific point, certain results appear stable and predictable.
Understanding limits fuels comprehension of how functions behave not just at points, but around them too. Some insights around limits and continuity include:

• If function outputs for \( h \) near zero stabilize at a number, it suggests continuity at that point.
• This concept ensures you know if sudden jumps or breaks exist in the function.
• Mastering limits equips students to deal with more complex calculus theories.

Overall, limits and continuity are like a sneak peek into a function's behavior near specific values in calculus.