Question

T/F: If \(\lim _{x \rightarrow 5} f(x)=\infty\), then \(f\) has a vertical asymptote at \(x=5\)

Short answer

True, \( f \) has a vertical asymptote at \( x=5 \).

Step-by-step solution

Understanding the Limit Notation

The notation \( \lim _{x \rightarrow 5} f(x)=\infty \) signifies that as \( x \) approaches 5, the function \( f(x) \) increases or decreases without bound. This means the function doesn't settle into a particular value but moves towards infinity or negative infinity.

Definition of a Vertical Asymptote

A vertical asymptote at \( x = c \) occurs when \( \lim_{x \to c^+} f(x) = \pm \infty \) or \( \lim_{x \to c^-} f(x) = \pm \infty \). This suggests that as \( x \) nears \( c \), \( f(x) \) becomes infinitely large or infinitely small.

Evaluating the Condition

Given that \( \lim _{x \rightarrow 5} f(x)=\infty \), it implies that as \( x \) approaches 5, \( f(x) \) behaves unboundedly. This directly aligns with the definition of a vertical asymptote, as \( f(x) \) tends towards infinity or negative infinity when approaching 5.

Final Conclusion

Since the condition provided exactly matches the criterion necessary for a vertical asymptote (i.e., the function heading to infinity), \( f(x) \) indeed has a vertical asymptote at \( x = 5 \). Therefore, the statement is True.

Key concepts

Limits

In calculus, a limit describes the behavior of a function as its input value approaches a particular point. It provides a way to understand what pattern or trend the function follows. A common scenario is understanding how a function behaves when the input value gets very close to a specific number.

In the provided exercise, where \( \lim _{x \rightarrow 5} f(x)=\infty \), the notation indicates that as \( x \) gets closer to 5, the output \( f(x) \) becomes infinitely large. This is crucial because it tells us that the function doesn't settle into a normal number as \( x \) approaches a particular value. Instead, the output shoots up to infinity. Not all functions have such behavior as \( x \) approaches a value. This unique characteristic hints towards an asymptotic behavior.

Infinity

Infinity is a concept in mathematics that represents something that is unbounded or unlimited. It's not a number, but an idea of a value that is larger than any real number.

When we say that the limit of \( f(x) \) as \( x \rightarrow 5 \) is infinity, it implies that the values of \( f(x) \) increase beyond any finite bound as \( x \) nears 5. The function does not stabilize; it continues to grow without limit. This concept of approaching infinity is key in determining the presence of vertical asymptotes in functions.

Vertical asymptotes themselves suggest that no matter how close we zoom into the function, as we approach a specific x-value, the function will shoot off to infinity or neged infinity.

Function Behavior as x Approaches a Value

Understanding how a function behaves as the input variable \( x \) approaches a particular value is significant in calculus. It helps us identify certain features of the function, like vertical asymptotes.

When considering the example \( \lim _{x \rightarrow 5} f(x)=\infty \), it suggests that this function, as \( x \) nears 5, does not settle at a particular number but keeps getting larger and larger.

• As \( x \) approaches from the left or the right, the function \( f(x) \) moves towards infinity.
• This trend shows us that there's no horizontal stabilizing factor at this point.

Such behavior is typically indicative of a vertical asymptote. The function essentially goes off the charts around \( x = 5 \).Understanding this allows us to predict the behavior of the function in this vicinity, as well as apply this knowledge to similar problems it might encounter in calculus.