A hole indicates that the function is not defined at that x-value, but the limit may still exist if the function approaches a specific y-value.

A vertical asymptote indicates that the limit approaches infinity (or negative infinity) as x approaches a certain value from one or both sides.

It suggests that the limit does not exist because the function does not approach a single value.

Examine the behavior of the function as x approaches the value from the left and from the right separately.

The graph must not have any breaks, holes, or jumps at that point. You should be able to draw the graph through that point without lifting your pen.

You can see the function being 'squeezed' between two other functions that converge to the same limit.

The graph visually shows that as x approaches 0, the function approaches 1, even though the function is undefined at x = 0.

The slope of the tangent line at a point is the limit of the difference quotient as h approaches 0, which represents the derivative at that point.

Look for vertical asymptotes. If the function approaches infinity (or negative infinity) as x approaches a certain value, then the limit is infinite.

The limit does not exist at a jump discontinuity because the left-hand limit and the right-hand limit are not equal.

1. Locate 'a' on the x-axis. 2. Follow the graph as x approaches 'a' from both sides. 3. Identify the y-value that the graph approaches. 4. That y-value is the limit.

1. Look at x-values approaching 'a' from both sides. 2. Observe the corresponding y-values. 3. If y-values approach the same number, that's the limit.

1. Try to factor and simplify the expression. 2. If simplification doesn't work, consider L'Hôpital's Rule. 3. Take the derivative of the numerator and denominator. 4. Evaluate the limit again.

1. Multiply the numerator and denominator by the conjugate of the expression containing the radical. 2. Simplify the expression. 3. Evaluate the limit.

1. Divide both numerator and denominator by the highest power of x in the denominator. 2. Evaluate the limit as x approaches infinity. Terms like 1/x approach 0.

1. Find the limit from the left. 2. Find the limit from the right. 3. If the left-hand limit equals the right-hand limit, the limit exists and is equal to that value.

1. Find two functions, f(x) and h(x), such that f(x) <= g(x) <= h(x). 2. Find the limits of f(x) and h(x) as x approaches a. 3. If both limits are equal to L, then the limit of g(x) as x approaches a is also L.

1. Find $\lim_{x \to a} g(x) = L$. 2. Find $\lim_{x \to L} f(x)$. 3. If this limit exists, it is the limit of the composite function.

1. Determine which piece of the function applies as x approaches the target value. 2. Evaluate the limit using that piece of the function. 3. If the target value is the boundary, check both left and right limits.

1. First, try direct substitution. 2. If direct substitution results in an indeterminate form, then use algebraic manipulation techniques.

Direct Substitution: Plug in the value directly. | Algebraic Manipulation: Simplify the expression before plugging in the value (factoring, conjugates, etc.).

Graph: Visual representation of the function's behavior. | Table: Numerical representation of the function's values.

L'Hôpital's Rule: Used for indeterminate forms by taking derivatives. | Squeeze Theorem: Used to find limits by bounding a function between two others.

One-sided limits: Approach from either the left or the right. | Two-sided limits: Approach from both sides, and both must agree for the limit to exist.

Limit exists: The function approaches a value. | Continuous: The limit exists, the function is defined, and they are equal.

Removable: Can be 'fixed' by redefining the function at a single point. | Non-removable: Cannot be fixed (e.g., jump, asymptote).

Finite values: Focus on the function's behavior near a specific point. | Infinity: Focus on the function's end behavior as x grows without bound.

Limit Laws: Apply basic properties of limits (sum, product, etc.). | Algebraic Manipulation: Transform the expression to simplify it before applying limit laws.

0/0: Both numerator and denominator approach zero. | ∞/∞: Both numerator and denominator approach infinity.

Conjugates: Used for expressions with radicals. | Factoring: Used for polynomial expressions.