1.14 Connecting Infinite Limits and Vertical Asymptotes

• What is a vertical asymptote?

  Line function approaches
• Infinite limits are mathematically represented as $\lim_{x \to a} f(x) =$$\pm \infty$, where $\infty$ denotes positive or negative infinity
• Match the concept with its definition:

  Infinite Limit ↔️ Limit approaches positive or negative infinity

  Vertical Asymptote ↔️ Vertical line the function approaches
• What happens to $f(x) =$$\frac{1}{x - 3}$ as $x$ approaches 3 from the left?


  $f(x) \to - \infty$
• An infinite limit occurs when a function's values approach infinity as the input variable approaches a certain value.


  True
• What is the mathematical representation of an infinite limit?

  \lim_{x \to a} f(x) = \pm \infty</latex>
• Match the concept with its definition:

  Infinite Limit ↔️ Values approach infinity

  Vertical Asymptote ↔️ Line the function approaches
• The presence of an infinite limit at $x =$$a$ indicates a vertical asymptote at $x =$$a$
• As $x$ approaches 3 from the right in f(x) = \frac{1}{x - 3}</latex>, $f(x)$ approaches $\infty$


  True
• What condition must be met for $x =$$a$ to be a vertical asymptote?


  $\lim_{x \to a} f(x) =$$\pm \infty$
• For $f(x) =$$\frac{1}{x - 2}$, where is the denominator zero?


  $x =$$2$
• What is an infinite limit?

  Limit approaches infinity
• When do infinite limits occur?

  Values approach infinity
• What is a vertical asymptote in terms of function behavior?

  A line function approaches
• If a function has an infinite limit at $x =$$a$, it also has a vertical asymptote at $x =$$a$.


  True
• The presence of an infinite limit at $x =$$a$ indicates a vertical asymptote at x = a
• Steps to identify vertical asymptotes using infinite limits:

  1️⃣ Identify potential locations where the denominator is zero

  2️⃣ Check for infinite limits as x approaches these values

  3️⃣ Confirm vertical asymptotes if $\lim_{x \to a} f(x) =$$\pm \infty$
• Vertical asymptotes are vertical lines that a function crosses but never touches.

  False
• What is an infinite limit in calculus?

  Limit approaches infinity
• What is a vertical asymptote?

  Line function approaches but never touches
• Steps to identify vertical asymptotes from infinite limits

  1️⃣ Find values of $x$ where the denominator of $f(x)$ is zero

  2️⃣ Evaluate limits as $x$ approaches these values from both left and right

  3️⃣ If $\lim_{x \to a} f(x) =$$\pm \infty$, then $x =$$a$ is a vertical asymptote
• To confirm a vertical asymptote, the limit as $x$ approaches $a$ must equal $\pm \infty$
• Match the concept with its definition:

  Infinite Limit ↔️ Limit approaches infinity

  Vertical Asymptote ↔️ Line function approaches but never touches
• Vertical asymptotes can be identified by analyzing infinite limits
• Vertical asymptotes occur when a function has a vertical limit of positive or negative infinity at a specific input value.


  True
• Steps to find vertical asymptotes

  1️⃣ Identify values where the denominator is zero

  2️⃣ Check if the limit as $x$ approaches those values is infinite
• What is a vertical asymptote?

  Line the function approaches
• Match the concept with its definition:

  Infinite Limit ↔️ Limit approaches infinity

  Vertical Asymptote ↔️ Line the function approaches
• Vertical asymptotes occur when the function's value approaches infinity as x approaches a certain value
• How are infinite limits and vertical asymptotes connected?

  Infinite limits indicate vertical asymptotes
• As $x$ approaches 3 from the left in $f(x) =$$\frac{1}{x - 3}$, $f(x)$ approaches $- \infty$


  True
• The presence of an infinite limit at $x =$$a$ indicates a vertical asymptote at $x =$$a$
• The denominator of f(x) = \frac{1}{x - 4}</latex> is zero at $x =$$4$


  True
• The function $f(x) =$$\frac{1}{x - 2}$ has a vertical asymptote at $x =$$2$


  True