Problem 58 - Precalculus with Limits

Question

In Exercises 49-68, find the limit by direct substitution. $$ \lim_{x \to 3}\ \dfrac{x^2+1}{x}$$

Step-by-Step Solution

Answer

The limit of the function \( \dfrac{x^2+1}{x} \) as \( x \) approaches 3 is \( \dfrac{10}{3} \).

Step 1: Direct Substitution

Directly substitute the value 3 for \( x \) in the given equation, obtaining \( \dfrac{3^2+1}{3} \).

Step 2: Simplify the Numerator

Simplify the square in the numerator, yielding \( \dfrac{9+1}{3} \).

Step 3: Perform the Addition

Compute the addition in the numerator, giving \( \dfrac{10}{3} \).

Step 4: Simplify the Fraction

Simplify the fraction to obtain a decimal or mixed number if necessary. In this case, the result \( \dfrac{10}{3} \) can be kept as it is because it can't be simplified further.

Key Concepts

Direct Substitution Method

When it comes to finding the limit of a function as the variable approaches a particular value, one of the simplest approaches is the direct substitution method. This technique involves replacing the variable in the function with the value it's approaching.

For example, if we have the mathematical expression \( \lim_{x \to a} f(x) \) and want to find the limit as x approaches a, we directly substitute x with a in the function f(x). Provided the function is continuous at that point, and no division by zero or other undefined behavior occurs, this straightforward substitution will yield the limit.

In the exercise provided, the limit of \( \frac{x^2+1}{x} \) as x approaches 3 is found using direct substitution. By replacing every occurrence of x with 3, we can instantly evaluate the expression without complex manipulations, revealing the limit with ease.

Simplifying Mathematical Expressions

Simplifying mathematical expressions is crucial for working efficiently with algebraic equations, solving for variables, and, as in this context, finding limits in calculus. To simplify an expression means to rewrite it in its most basic form without changing its value.

This usually involves several steps, such as distributing products over sums, combining like terms, reducing fractions, and factoring.

• For instance, squaring binomials, condensing exponents, and combining constant terms can help to present a clearer and more concise form of the expression.
• You might also look for common factors in numerators and denominators to cancel terms out when simplifying fractions.
• Ultimately, the goal is to make the problem easier to understand and solve.

In our exercise, the given function is simplified by squaring the number 3 and adding to 1, leading to a simple fraction that shows the limit clearly as \( \frac{10}{3} \) after the arithmetic is completed.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding continuity, derivatives, and the behavior of functions on an intuitive level.

Formally, we write \( \lim_{x \to a} f(x) = L \) to mean that as x gets very close to a, f(x) gets very close to the number L. This L is the limit we seek. It's important to note that x does not need to be equal to a for the limit to exist, nor does the function need to be defined at a.

Common Scenarios Involving Limits

• If the function approaches the same value from both the left and right as x approaches a particular number, then the limit exists.
• When the values lead to an undefined situation, such as division by zero, other techniques than direct substitution must be used.
• Limits also come into play with infinite sequences and series, and they help describe the behavior of functions at infinity.

Our example problem simply required direct substitution to find that the limit is \( \frac{10}{3} \) as x approaches 3.