Graph each function and then find the specified limits. When...

Graph each function and then find the specified limits. When necessary, state that the limit does not exist.

$$f(x)=\frac{1}{x}+3 ; \text { find } \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow 0} f(x).$$

Key Concepts

Limits of Functions

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a particular point or infinity. They help to determine the value that a function tends to, even if the function is not defined at that point, and form the basis for further topics like continuity, differentiation, and integration.

Horizontal Asymptotes

A horizontal asymptote represents the value that a function approaches as the independent variable goes to positive or negative infinity. This concept is pivotal when analyzing the end behavior of functions, particularly rational functions, where the degrees of the numerator and denominator determine the asymptotic value.

Vertical Asymptotes

Vertical asymptotes occur where a function grows without bound or drops to negative infinity as the input approaches a certain finite value, often due to division by zero or discontinuities in the function. This concept is crucial in understanding the local behavior near points of discontinuity.

Reciprocal Functions

Reciprocal functions, such as those of the form 1/x, exhibit unique characteristics including a vertical asymptote where the function is undefined and a horizontal asymptote reflecting the function’s behavior at infinity. Understanding these functions enhances comprehension of general function transformations and asymptotic behavior.

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