Question

Find the limit. $$ \lim _{x \rightarrow 7} \frac{5 x}{x+2} $$

Short answer

\(\frac{35}{9}\)

Step-by-step solution

Identify the limiting value

The limiting value is given in the problem as \(x \rightarrow 7\).

Use the method of substitution

Substitute \(x = 7\) into the function \(\frac{5x}{x+2}\). This gives \(\frac{5 * 7}{7 + 2} = \frac{35}{9}\).

Provide the solution

The limit of the function as \(x\) approaches 7 is \(\frac{35}{9}\).

Key concepts

Substitution Method

The substitution method is a fundamental technique in calculus used to evaluate limits. When we say we use substitution, we mean that we replace the variable with the value it is approaching. In the context of limits, this often involves substituting the given value of the variable directly into the function. Doing so helps us determine the behavior of the function as the variable nears a specific point.

• Example: If you have a function and you're asked to find the limit as \(x\) approaches a certain number, you substitute this number into the function to see what value the function approaches.
• Practical step: If direct substitution does not work (for instance, if it results in an undefined expression like 0/0), further techniques such as algebraic simplification or L'Hôpital's rule might be needed.

In this exercise, we applied the substitution method by replacing \(x\) with 7 in the rational function \(\frac{5x}{x+2}\), which led us to the simplified expression and ultimately the limit: \(\frac{35}{9}\).

Rational Functions

A rational function is a type of function represented by a fraction where both the numerator and the denominator are polynomials. These functions are common in calculus and are quite important when discussing limits.

• General Form: A rational function can be expressed as \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials.
• Behavior: The behavior of rational functions can often be unpredictable as \(x\) approaches certain values, especially at points where the denominator becomes zero.
• Simplification: Simplifying the function or the expression can sometimes resolve issues where direct substitution into the rational function is problematic.

In this particular problem, we dealt with a rational function: \(\frac{5x}{x+2}\). The simplicity of this function allowed us to directly substitute \(x = 7\) without any additional algebraic manipulation as the denominator \(7+2\) does not approach zero.

Limits and Continuity

Limits and continuity are key concepts in the study of calculus. The concept of a limit describes the value a function approaches as the input approaches a certain value. It helps us understand how functions behave at points they might not be directly evaluated at due to indeterminate forms or discontinuities. Continuity, on the other hand, describes a function that is unbroken and smoothly connected over its domain.

• Understanding Limits: Limits allow us to explore the behavior of a function at points where direct evaluation might not give a clear answer.
• Function Behavior: They can show how a function behaves as \(x\) approaches a particular value, even if not directly accessible due to discontinuities.
• Continuity Implications: A function is continuous at a point if its limit as \(x\) approaches the point gives the same result as directly substituting the point into the function.

In our example, finding the limit of \(\frac{5x}{x+2}\) as \(x\) approaches 7 shows us the value the function converges to at this point. The fact that this limit works smoothly, showing \(\frac{35}{9}\), suggests there is continuity at \(x = 7\), emphasizing that the substitution method applies easily without special considerations for discontinuities.