if the line $y = a$ is a vertical asymptote of the function $f(x)$, then at least one of the following…

if the line $y = a$ is a vertical asymptote of the function $f(x)$, then at least one of the following statements is true select all that apply.\n$lim_{x\rightarrow a}f(x)=pminfty$\n$lim_{x\rightarrow a^{-}}f(x)=pminfty$\n$lim_{x\rightarrow a^{+}}f(x)=pminfty$\n$lim_{x\rightarrow a^{+}}f(x)=a$\n$lim_{x\rightarrow a^{-}}f(x)=a$

if the line $y = a$ is a vertical asymptote of the function $f(x)$, then at least one of the following statements is true select all that apply.\n$lim_{x\rightarrow a}f(x)=pminfty$\n$lim_{x\rightarrow a^{-}}f(x)=pminfty$\n$lim_{x\rightarrow a^{+}}f(x)=pminfty$\n$lim_{x\rightarrow a^{+}}f(x)=a$\n$lim_{x\rightarrow a^{-}}f(x)=a$

Answer

Explanation:

Step1: Recall vertical - asymptote definition

The line $x = a$ is a vertical asymptote of the function $y = f(x)$ if either the left - hand limit $\lim_{x\rightarrow a^{-}}f(x)=\pm\infty$, the right - hand limit $\lim_{x\rightarrow a^{+}}f(x)=\pm\infty$, or the two - sided limit $\lim_{x\rightarrow a}f(x)=\pm\infty$.

Step2: Analyze each option

  • Option 1: $\lim_{x\rightarrow a}f(x)=\pm\infty$ implies that as $x$ approaches $a$ from both sides, $f(x)$ goes to positive or negative infinity. This is a valid condition for $x = a$ to be a vertical asymptote.
  • Option 2: $\lim_{x\rightarrow a^{-}}f(x)=\pm\infty$ means that as $x$ approaches $a$ from the left - hand side, $f(x)$ goes to positive or negative infinity. This is a valid condition for $x = a$ to be a vertical asymptote.
  • Option 3: $\lim_{x\rightarrow a^{+}}f(x)=\pm\infty$ implies that as $x$ approaches $a$ from the right - hand side, $f(x)$ goes to positive or negative infinity. This is a valid condition for $x = a$ to be a vertical asymptote.
  • Option 4: $\lim_{x\rightarrow a^{+}}f(x)=a$ is not a condition for a vertical asymptote. For a vertical asymptote, the function values should be going to $\pm\infty$ as $x$ approaches $a$, not a finite value $a$.
  • Option 5: $\lim_{x\rightarrow a^{-}}f(x)=a$ is not a condition for a vertical asymptote. For a vertical asymptote, the function values should be going to $\pm\infty$ as $x$ approaches $a$, not a finite value $a$.

Answer:

$\lim_{x\rightarrow a}f(x)=\pm\infty$, $\lim_{x\rightarrow a^{-}}f(x)=\pm\infty$, $\lim_{x\rightarrow a^{+}}f(x)=\pm\infty$