Problem 59 - Thomas Calculus

Question

Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form $$\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$ occur frequently in calculus. In Exercises \(57-62,\) evaluate this limit for the given value of \(x\) and function \(f\) . $$f(x)=3 x-4, \quad x=2$$

Step-by-Step Solution

Answer

The value of the limit is 3.

Step 1: Substitute the function and x-value

First, substitute the given function, \(f(x) = 3x - 4\), and the x-value, \(x = 2\), into the limit expression given by: \[ \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \] This leads to \[ \lim _{h \rightarrow 0} \frac{(3(x+h) - 4) - (3x - 4)}{h} \] With \(x = 2\), the expression becomes: \[ \lim _{h \rightarrow 0} \frac{(3(2+h) - 4) - (3(2) - 4)}{h} \]

Step 2: Simplify the expression inside the limit

Simplify the expression inside the limit: Start by expanding and simplifying the terms inside the numerator: \[ \lim _{h \rightarrow 0} \frac{(6 + 3h - 4) - (6 - 4)}{h} \] This simplifies to: \[ \lim _{h \rightarrow 0} \frac{2 + 3h - 2}{h} \] Which further simplifies to: \[ \lim _{h \rightarrow 0} \frac{3h}{h} \]

Step 3: Evaluate the limit

Cancel \(h\) in the numerator and denominator, provided \(h eq 0\). The expression becomes: \[ \lim _{h \rightarrow 0} 3 \] Since the limit no longer depends on \(h\), evaluate it by direct substitution: \[ 3 \] This is the value of the limit.

Key Concepts

Limits

Limits are a cornerstone concept in calculus that help us understand behavior as values approach a specific point. In this context, limits allow us to analyze how a function behaves as it nears a particular value, which is essential in determining derivatives and understanding rates of change.
When dealing with the expression \( \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \), we're working with the limit that defines the derivative. It essentially represents the slope of the tangent line at a particular point on the function's graph. The goal is to evaluate the limit by observing what value the function approaches as \( h \) gets infinitely close to 0.
In our example, the simplification of the limit involves canceling terms to get \( \frac{3h}{h} \), where the \( h \) terms can be canceled out since \( h eq 0 \). After canceling, the expression simplifies directly to 3. Thus, the limit is determined to be 3 as \( h \) approaches 0.

Tangent Lines

Tangent lines play a vital role in calculus, particularly in understanding the geometry of curves. A tangent line at any given point runs in a direction that just "touches" the curve at that point, without crossing it.
The slope of a tangent line is what we derive using limits, specifically using the expression \( \lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h} \). This expression is the definition of a derivative, and when evaluated, it gives us the slope of the tangent line.

• When a derivative is computed, it essentially gives the slope of the tangent line at a point \( x \).
• For linear functions, like \( f(x) = 3x - 4 \), the derivative comes out as a constant, because the slope does not change.

In our example, since the line is straight, the tangent at any point is simply the line itself, meaning the slope of the tangent line is the same at all points, which we've calculated as a constant value of 3.

Calculus

Calculus, a branch of mathematics, is fundamentally concerned with change and motion. It provides us with tools to analyze how things evolve over time, covering a wide variety of topics like velocity, rates of change, and area under curves.
Two primary tools in calculus are derivatives and integrals:

• Derivatives: They provide information about the rate of change of quantities. This is what we calculate using limits to find slopes of tangent lines.
• Integrals: Often thought of as the inverse process of differentiation, they are used to calculate the total accumulation of quantities, like area.

In calculus, the concept of a derivative, which we've applied in this problem, allows us to derive the slope function of any given point on a curve. It is incredibly powerful in predicting movement and growth patterns. This utility extends beyond theoretical exercises, underpinning much of the modern scientific and technological world by providing insights into complexities of natural and engineered systems.