1. Factor numerator and denominator. 2. Simplify the function. 3. Find values where the denominator is zero and the numerator is non-zero.

1. Evaluate $\lim_{x\to a^+} f(x)$. 2. Evaluate $\lim_{x\to a^-} f(x)$. 3. Show that at least one of these limits is $\pm \infty$.

1. Factor and simplify the function. 2. If a factor cancels, it's a removable discontinuity. 3. If a factor remains in the denominator, it's a vertical asymptote.

1. Factor: $f(x) = \frac{x+2}{(x+2)(x-2)}$. 2. Simplify: $f(x) = \frac{1}{x-2}$. 3. VA at $x=2$.

1. Rewrite as $f(x) = \frac{\sin(x)}{\cos(x)}$. 2. Find where $\cos(x) = 0$. 3. $x = \frac{(2n+1)\pi}{2}$, where n is an integer.

1. Factor: $f(x) = \frac{(x-1)(x+1)}{x-1}$. 2. Simplify: $f(x) = x+1$. 3. Removable discontinuity at x=1, no vertical asymptote.

1. Determine the sign of the function as x approaches from the left and right. 2. The limit will be either positive or negative infinity.

1. Set denominator equal to zero: $x^2 + 1 = 0$. 2. Solve for x: $x^2 = -1$. 3. No real solutions, so no vertical asymptotes.

1. Check where the denominator is zero: $x=0$. 2. Verify the numerator is not zero at $x=0$: $e^0 = 1 \neq 0$. 3. Vertical asymptote at $x=0$.

1. Check limit from the right: $\lim_{x\to 2^+} \frac{1}{x-2} = \infty$. 2. Check limit from the left: $\lim_{x\to 2^-} \frac{1}{x-2} = -\infty$. 3. Vertical asymptote at $x=2$.

If the limit of a function as $x$ approaches $a$ is infinite, then $x=a$ is a vertical asymptote.

They indicate points where a function is undefined and help visualize the function's behavior, especially concerning limits.

To understand how the function behaves on both sides of the asymptote and confirm its existence.

As $x$ approaches $a$ from the right, $f(x)$ approaches positive infinity. As $x$ approaches $a$ from the left, $f(x)$ approaches negative infinity.

Because $\lim_{x\to 0^+} ln(x) = -\infty$, and $ln(x)$ is undefined for $x \le 0$.

Factoring can simplify rational functions, revealing removable discontinuities and true vertical asymptotes.

A vertical asymptote occurs where the function approaches infinity, while a removable discontinuity is a hole in the graph that can be 'removed' by redefining the function.

A positive infinite limit means the function increases without bound, while a negative infinite limit means it decreases without bound.

Simplifying can reveal common factors in the numerator and denominator, which indicate removable discontinuities rather than vertical asymptotes.

The graph will approach the vertical asymptote very closely, with the y-values either increasing without bound (approaching positive infinity) or decreasing without bound (approaching negative infinity).

A vertical line $x=a$ that a function approaches but never touches; function is unbounded at $x=a$.

A point where a function is undefined or behaves erratically.

Limits that evaluate to either positive or negative infinity.

The function's value grows or shrinks without any bound, approaching infinity or negative infinity.

A vertical asymptote is a type of discontinuity where the function is undefined.

As $x$ approaches $a$, the value of $f(x)$ increases without bound.

As $x$ approaches $a$, the value of $f(x)$ decreases without bound.

A limit that considers the function's behavior as x approaches a value from either the left or the right.

The limit of $f(x)$ as $x$ approaches $a$ from the right.

The limit of $f(x)$ as $x$ approaches $a$ from the left.