Question

Finding a function with infinite limits Give a formula for a function \(f\) that satisfies \(\lim _{x \rightarrow 6^{+}} f(x)=\infty\) and \(\lim _{x \rightarrow 6^{-}} f(x)=-\infty\)

Short answer

Answer: The function that meets the given requirements is \(f(x)=\frac{b}{x-6}\), where \(b\) is a nonzero constant.

Step-by-step solution

Check limit as x approaches 6 from the positive side (6+)

Calculate the limit: \(\lim _{x \rightarrow 6^{+}} \frac{b}{x-6}\) Since the numerator is a constant and as x approaches 6 from the positive side, the denominator \((x-6)\) approaches 0 from the positive side, making the expression go towards infinity. Hence, the condition \(\lim _{x \rightarrow 6^{+}} f(x)=\infty\) is satisfied. 2. Now, let's check for the limit when x approaches 6 from the negative side (\(6^-\)).

Check limit as x approaches 6 from the negative side (6-)

Calculate the limit: \(\lim _{x \rightarrow 6^{-}} \frac{b}{x-6}\) Similarly, as x approaches 6 from the negative side, the denominator \((x-6)\) approaches 0 from the negative side, making the expression go towards negative infinity. Hence, the condition \(\lim _{x \rightarrow 6^{-}} f(x)=-\infty\) is satisfied. As both conditions are satisfied, we can conclude that \(f(x)=\frac{b}{x-6}\), where \(b\) is a nonzero constant, is the function that meets the given requirements.

Key concepts

Infinite Limits

When we talk about infinite limits, we refer to the behavior of a function as the input, or variable, approaches a particular value resulting in the output growing without bound. Imagine you have a number line that stretches onwards forever in both directions. When a function's output heads towards infinity, it means that as the input gets closer and closer to a certain value, the output climbs higher and higher, without ever settling at a particular number.

In the exercise, as we approach the number 6 from the right (or positive side), our function shoots up to infinity. This happens because the denominator (\( x-6 \)) gets very small and positive, thereby making the overall fraction larger and larger as it approaches zero. This is the essence of infinite limits: an input nears a specific point causing the output to grow endlessly.

• Infinite limits involve outputs that increase or decrease without bound.
• They occur as the input comes very close to a certain value but never really touches it.
• These limits illustrate how functions can behave as variables approach specific numbers.

One-sided Limits

One-sided limits are a crucial concept in calculus. They look at the behavior of a function as the input comes towards a specific point, focusing exclusively from one side - either from the left (negative direction) or the right (positive direction).

In the original exercise, we checked both \( \lim_{x \rightarrow 6^{+}} f(x) \)and\( \lim_{x \rightarrow 6^{-}} f(x) \). This means we assessed the function's limit from both the right and the left of 6:

• From the right side (\( 6^{+} \)), where the function tends towards positive infinity.
• From the left side (\( 6^{-} \)), where it runs toward negative infinity.

One-sided limits help in understanding the complete behavior of functions near specific values, which can vastly differ based on approach direction. They are helpful to graph and predict behaviors of crucial functions including but not limited to rational functions.

Functions Approaching Infinity

In mathematics, when we discuss functions approaching infinity, we're describing a scenario where, as the input gets closer to a certain value, the function's output increases or decreases without end. Functions exhibit this behavior through their algebraic structure or the way they're graphed.

Our given function, \( f(x) = \frac{b}{x-6} \), is an excellent example of this. As \( x \) draws near to 6, the numerator remains constant, but the denominator nears zero, causing the function to stretch towards infinity and negative infinity depending on the side from which \( x \) approaches. This behavior is critical in understanding rational functions, where denominators approaching zero lead to steep increases or decreases in the function values.

Functions that approach infinity are significant in numerous scientific and engineering contexts, where predictions and models involve extreme values. By understanding how functions behave as they reach these extreme limits, mathematicians and scientists can better grasp real-world phenomena and system behaviors.