Question

Finding a Limit In Exercises \(41-46,\) write a simpler function that agrees with the given function at all but one point. Then find the limit of the function. Use a graphing utility to confirm your result. $$ \lim _{x \rightarrow 0} \frac{x^{2}+3 x}{x} $$

Short answer

The limit as \(x\) approaches 0 of the function \(\frac{x^{2}+3 x}{x}\) equals 3.

Step-by-step solution

Simplification

To simplify the function, divide each term by \(x\). \(\frac{x^{2}}{x} + \frac{3x}{x} = x + 3\) is hence the simplified function.

Calculate the Limit

With the simplified function \(x + 3\), as \(x\) approaches 0, it results in \(0 + 3 = 3\).

Verification

You can verify the limit by plugging in values close to 0 into the simplified function. Both negative and positive values that are close to 0 should result in the limit of 3.

Key concepts

Calculating Limits

Understanding how to calculate limits is an essential skill in calculus. A limit represents the value a function approaches as the input (or 'x' value) gets closer to a certain number. When we look at the limit of \( \frac{x^{2}+3x}{x} \) as \( x \) approaches 0, we're trying to find out what value the function is getting closer to when \( x \) is near to 0.

To accurately calculate limits without encountering undefined expressions, simplification is often necessary. In our exercise, simplifying the function \( \frac{x^{2}+3x}{x} \) by dividing each term in the numerator by \( x \) leads to \( x + 3 \) because \( \frac{x^{2}}{x} = x \) and \( \frac{3x}{x} = 3 \) by the properties of division. This simplification step avoids the division by zero issue which would occur if we directly substituted \( x \) with 0.

Upon obtaining the simplified function, we calculate the limit by substituting \( x \) with 0 directly into \( x + 3 \) which gives us 3. Hence, \( \lim\text{ }\frac{x^{2}+3x}{x} \rightarrow 0 \) is 3. A crucial aspect to remember is that even if the original function was not defined at \( x = 0 \) due to division by zero, the limit can still exist at that point and be different from the function's value.

Simplifying Functions

Simplifying functions often makes finding limits easier and helps to reveal the overall behavior of a function. Simplification can involve factoring, expanding, canceling common factors, and other algebraic manipulations that reduce the complexity of a function.

For example, in the exercise we're discussing, the original function \( \frac{x^{2}+3x}{x} \) was simplified to \( x + 3 \) by canceling the common factor of \( x \) in the numerator and the denominator. This is possible whenever \( x eq 0 \) since division by zero is undefined. Simplification not only aids in calculating limits but also in graphing the function and understanding its behavior across different values of \( x \).

When simplifying, it's vital to check if the simplification changes the domain of the function or its overall shape. It’s worth noting that the simplified function \( x + 3 \) is equivalent to the original function at every point except \( x = 0 \) and this exception is what allows for the calculation of the limit as \( x \) approaches 0.

Graphing Utilities in Calculus

Graphing utilities, such as calculators and software programs, are invaluable tools in calculus. They allow students to visualize functions and their behavior, especially near points of interest for limits.

In our exercise, using a graphing utility would help confirm the limit of \( \frac{x^{2}+3x}{x} \) as \( x \) approaches 0 by plotting the graph and observing the y-value as \( x \) values get closer and closer to 0. If the graph approaches a certain y-value, then that would be the limit that we have previously calculated algebraically.

Graphing utilities also help in understanding concepts like continuity and asymptotes, which are critical in calculus. Observing the graph of \( x + 3 \) can confirm its linearity and that it indeed passes through the point (0, 3), reinforcing our calculated limit. Seeing the graph can make the abstract concept of limits more tangible and can help bridge the gap between numerical calculation and visual understanding.