Question

Find the limit \(L .\) Then find \(\delta>0\) such that \(|f(x)-L|<0.01\) whenever \(0<|x-c|<\delta\). $$ \lim _{x \rightarrow 4}\left(4-\frac{x}{2}\right) $$

Short answer

The limit \( L \) is 2, and the delta \( \delta \) is 0.02

Step-by-step solution

Calculate the Limit

To start, first calculate the limit of the function as x approaches 4. The function is linear, so simply substitute x with 4 to get \( L = 4 - \frac{4}{2} = 2 \). Hence, the limit \( L \) is 2.

Find the Delta

Next, find a delta such that the absolute difference between the function \( f(x) = 4 - x/2 \) and the limit \( L = 2 \) is less than 0.01 for \(0 < |x - c| < d\). To do that, set up the equation \( |f(x) - L| < 0.01 \), which becomes \( |4 - x/2 - 2| < 0.01 \), simplifies to \( |-x/2 + 2| < 0.01 \). The absolute value will be largest when \( x = 4 + d \) or \( x = 4 - d \). Create two inequalities based on this and solve for \( d \). The smaller \( d \) will be your \( \delta \).

Solve the inequalities

The two inequalities that are derived from the conditions are \( |-(4 + d)/2 + 2| < 0.01 \) and \( |-(4 - d)/2 + 2| < 0.01 \). After simple math, they simplify to \( d < 0.02 \) and \( d > -0.02 \). Thus the appropriate delta is the smaller positive solution , \( \delta = 0.02 \).

Key concepts

Limits in Calculus

A limit in calculus represents the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential for understanding the behavior of functions as they approach specific points or infinity.

They're the building blocks of calculus, allowing for the definition of derivatives and integrals. Calculus is often described as the study of change, and limits help quantify that change by looking at values that are not quite reached but are approached closer and closer.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as the input approaches a particular point. It's a way of talking about what happens to the function's value as we get infinitely close to a certain input value without actually reaching it.

For example, consider the limit statement \( \lim_{x \rightarrow a} f(x) = L \). It says that as x gets closer and closer to a, f(x) gets closer and closer to L. In the exercise provided, we found that as x approaches 4, the function \( f(x) = 4 - \frac{x}{2} \) gets closer and closer to the number 2.

Delta-Epsilon Definition

The delta-epsilon definition is a precise, mathematical way of defining what we mean by 'limit'. It uses two arbitrary positives, delta (\(\delta\)) and epsilon (\(\epsilon\)), to describe the behavior of a function near a certain point.

The formal definition states: For every \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < |x-c| < \delta\), it follows that \(|f(x) - L| < \epsilon\). Here, \(\epsilon\) represents how close the function values need to be to the limit, and \(\delta\) represents how close the input values need to be to the point of interest, c. By finding the appropriate delta in the exercise, we ensured that the function's values were within 0.01 of the limit when the x-values were sufficiently close to 4.

Continuity in Calculus

Continuity in calculus is about functions that are smooth and unbroken. A function is continuous at a point if there's no interruption in the graph at that point—technically, if it satisfies the three conditions: the function is defined at the point, the limit of the function exists at that point, and the limit matches the function's value.

If a function is continuous at every point in an interval, we just say it's continuous over that interval. The function in our exercise, \( f(x) = 4 - \frac{x}{2} \), is linear and therefore continuous at every point, including x = 4. This continuity guarantees that the delta-epsilon approach can be applied to find limits as in the solved exercise.