Limits and Asymptotes in Rational Functions

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Questions and Answers

What is the left limit of the function 𝑓(𝑥) as 𝑥 approaches 1 for the expression (𝑥 − 1)(𝑥 + 5)?

2 (correct)

−∞

What type of discontinuity is present at 𝑥 = 1 for the function 𝑓(𝑥) = (𝑥 − 1)(𝑥 + 2)?

Hole (correct)

Removable discontinuity

Jump discontinuity

Vertical asymptote

What is the left end behavior of the function $f(x)$ as $x$ approaches 1?

lim $f(x)$ = 1 as $x$ → 1 (correct)

lim $f(x)$ = 2 as $x$ → 1

lim $f(x)$ = 0 as $x$ → 1

lim $f(x)$ = -1 as $x$ → 1

What is the right limit of the function 𝑓(𝑥) = (𝑥 − 2)(𝑥 − 4) as 𝑥 approaches 1?

2 Signup and view all the answers

What is the right end behavior of the function $g(x)$ as $x$ approaches 2?

lim $g(x)$ = 2 as $x$ → 2 Signup and view all the answers

For the rational function 𝑓(𝑥) = −2(𝑥 + 3)(𝑥 + 1), what determines the behavior of the limit as 𝑥 approaches 1?

The values of both the numerator and denominator. Signup and view all the answers

What is the left end behavior of the function $k(x)$ as $x$ approaches 1?

lim $k(x)$ = 0 as $x$ → 1 Signup and view all the answers

As 𝑥 approaches 1 for 𝑓(𝑥) = (𝑥 − 1)(𝑥 + 2), what happens to the function's value?

Approaches 0 Signup and view all the answers

What is the right limit of 𝑓(𝑥) = −2(𝑥 + 3)(𝑥 + 1) as 𝑥 approaches 1?

-16 Signup and view all the answers

What is the right end behavior of the function $h(x)$ as $x$ approaches 3?

lim $h(x)$ = 0 as $x$ → 3 Signup and view all the answers

What can be concluded about the function 𝑓(𝑥) = (𝑥 − 1)(𝑥 + 2) at the point 𝑥 = 1?

It has a hole. Signup and view all the answers

What is the left end behavior of the function $r(x)$ as $x$ approaches 2?

lim $r(x)$ = 0 as $x$ → 2 Signup and view all the answers

What is the nature of the right limit of the function 𝑓(𝑥) as 𝑥 approaches 1 for (𝑥 − 2)(𝑥 − 4)?

Positive infinity Signup and view all the answers

What is the limit of 𝑦 as 𝑥 approaches 2?

5 Signup and view all the answers

Which value indicates a hole in the function 𝑘(𝑥)?

-2 Signup and view all the answers

What is the vertical asymptote of the function 𝑟(𝑥)?

-1 Signup and view all the answers

For which inequality does the solution set result in (−2,3]?

𝑥 − 1 ≤ 0 Signup and view all the answers

What is the limit of function $f(x)$ as the input values decrease without bound?

-∞ Signup and view all the answers

What is the limit of 𝑟(𝑥) as 𝑥 approaches -1?

−∞ Signup and view all the answers

Which of the following expressions indicates a vertical asymptote for 𝑦?

𝑥 + 3 = 0 Signup and view all the answers

What is the limit of function $g(x)$ as the input values increase without bound?

1 Signup and view all the answers

What is the limit of function $h(x)$ as the input approaches a specific value from the left?

-∞ Signup and view all the answers

When does the function 𝑘(𝑥) approach a limit of zero?

When 𝑥 approaches -1 Signup and view all the answers

For 𝑦 = (𝑥 + 3)/(𝑥 − 2), what value of 𝑥 creates a vertical asymptote?

2 Signup and view all the answers

What can be concluded about the limit of function $k(x)$ as the input approaches a specific value from the left?

-2 Signup and view all the answers

What is the limit of function $h(x)$ as the input approaches infinity?

∞ Signup and view all the answers

What is the limit of function $k(x)$ as the input approaches infinity?

∞ Signup and view all the answers

Which limit correctly describes the behavior of function $h(x)$ as the input approaches a specific value from the right?

3 Signup and view all the answers

What is the limit of function $k(x)$ when the input approaches the same specific value from both left and right?

3 Signup and view all the answers

What is the domain of the rational function represented by $\frac{x + 2}{(x + 1)}$?

All real numbers except $-1$ Signup and view all the answers

Which interval represents where the expression $x^2 - x - 12 \geq 0$ is true?

($-\infty$, $-3$) ∪ ($4$, $\infty$) Signup and view all the answers

Which of the following rational function expressions are in lowest terms?

$\frac{x - 3}{x^2 - 9}$ Signup and view all the answers

What is the range of the rational function $\frac{x + 2}{x + 1}$?

All real numbers except $1$ Signup and view all the answers

For the expression $-2x(x - 3)$, what are the critical points?

$0$ and $3$ Signup and view all the answers

Study Notes

Limits of Rational Functions

As (x) approaches 1, the left limit of (f(x) = \frac{(x - 1)(x + 5)}{(x - 2)(x - 4)}) evaluates to 2, while the right limit also evaluates to 2.
For (f(x) = \frac{(x - 1)(x + 2)}{(1 - 2)(1 - 4)}), the left limit approaches -∞ and the right limit approaches -∞.
The left limit of (f(x) = \frac{-2(x + 3)(x + 1)}{(x - 1)^2}) approaches -∞, and the right limit also approaches -∞.

Identifying Holes and Asymptotes

For (y = \frac{(x + 3)(x - 2)}{(x + 3)^2}), there is a hole at (x = -3) and a vertical asymptote at (x = -1).
For (k(x) = \frac{(x + 7)(x + 2)}{(x + 1)(x + 2)}), identify no hole due to the absence of common factors in numerator and denominator.
The vertical asymptote in (k(x)) is at (x = -1).
(r(x) = \frac{x^2 - x}{x^2(x - 1)}) has a hole at (x = 1) and a vertical asymptote at (x = -1).

Solving Inequalities

The inequality (x - 3 \leq 0) has solution ( (-\infty, 3] ).
The inequality ((x - 1)(x + 2) \geq 0) has solution ([-2, 1)).

Limit Behavior of Graphs

As input values decrease without bound for (f(x)), the limit approaches -∞.
For (g(x)), as input values increase without bound, the limit approaches 1.

Analyzing Graphs

Limit of (h(x)) approaches -∞ as (x) approaches values near the graph from the left and also tends to a finite value as input increases.
The function (k(x)) has a limit approaching -2 as (x) approaches a certain value from the left.

Summary of Function Behavior

(f(x)) and (g(x)) both stabilize at a limit of 1.
Horizontal asymptotes and behavior at infinity show functions approaching fixed value limits across specified intervals.
Continuous consideration of holes and vertical asymptotes is crucial for understanding graph behavior of rational functions.

Notable Functions and Their Limits

(f(x) = \frac{2x^2 - 2x + 1}{3x^2 + 5x + 7}) indicated a left limit of 1.
The graphs of (h(x)) and (k(x)) demonstrate distinct limits based on their structural forms and asymptotic behavior.

Analyzing Output Values

There are descriptions connecting input values to output limits, emphasizing continuity and variability in rational function graphs.

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Description

Test your understanding of limits, holes, and asymptotes in rational functions. This quiz includes problems on evaluating limits and identifying vertical asymptotes and holes in given rational expressions. Perfect for students studying calculus topics related to rational functions.