A limit describes the value that a function approaches as the input approaches some value. It focuses on the behavior near a point, not necessarily at the point itself.

One-sided limits examine the behavior of a function as it approaches a value from either the left (left-hand limit) or the right (right-hand limit).

A limit does not exist if the function is unbounded, oscillates wildly, or if the left-hand limit and right-hand limit are not equal.

By observing the y-value that the function approaches as x approaches a specific value. Trace the curve from both sides to see if they converge to the same y-value.

A jump discontinuity indicates that the left-hand limit and the right-hand limit are different, therefore the limit at that point does not exist.

A vertical asymptote indicates that the function is unbounded as x approaches a certain value, meaning the limit does not exist at that point.

For a function to be continuous at a point, the limit must exist at that point, the function must be defined at that point, and the limit must equal the function value.

The scale of a graph can distort the appearance of the function, potentially hiding important details or exaggerating certain behaviors, leading to incorrect limit estimations.

Checking one-sided limits is crucial because the overall limit exists only if both the left-hand limit and the right-hand limit exist and are equal.

Oscillation refers to a function fluctuating rapidly between two values as x approaches a specific point. If the oscillations become infinitely rapid, the limit does not exist.

Limit: Value the function approaches. | Function Value: Value the function is at that point.

Left-Hand: Approaching from the left. | Right-Hand: Approaching from the right.

Hole: Function undefined, limit may exist. | Vertical Asymptote: Function unbounded, limit DNE.

Jump: Left and right limits differ. | Removable: Limit exists, but doesn't equal the function value or function not defined.

Graphically: Visual estimation, potential for inaccuracy. | Algebraically: Precise calculation, requires function definition.

Continuous: Limit often equals the function value. | Discontinuous: Limit may or may not exist; requires careful examination.

One-sided: Function approaches a value from one direction. | Overall: Function approaches the same value from both directions.

Vertical Asymptote: Function values tend to infinity (or negative infinity). | Hole: Function is undefined, but values nearby are finite.

sin(x): Limit is 0. | sin(1/x): Limit DNE due to oscillation.

Polynomial: Limit always exists and is easily found by direct substitution. | Rational: Limit may or may not exist; check for asymptotes and holes.

1. Visualize the point. 2. Trace along the graph from both sides. 3. Check if the one-sided limits match.

1. Check for vertical asymptotes. 2. Check for jump discontinuities. 3. Check for wild oscillations.

Trace the graph from the left side of x=a. Determine the y-value the function approaches as x gets closer to a from the left.

Trace the graph from the right side of x=a. Determine the y-value the function approaches as x gets closer to a from the right.

Determine $$\lim_{x \to a^-} f(x)$$ and $$\lim_{x \to a^+} f(x)$$. If they are equal, that is the limit. Otherwise, the limit DNE.

The limit can still exist even with a hole. Focus on what y-value the function approaches as x approaches 'a' from both sides, not the value at x=a.

Check if the limit exists at x=a, if f(a) is defined, and if $$\lim_{x \to a} f(x) = f(a)$$.

Look for a point where the graph 'jumps' from one y-value to another. The left and right limits will be different at this point.

Look for a vertical line where the function approaches infinity (or negative infinity) as x approaches that line.

If the function oscillates wildly near a point, the limit likely does not exist. The function does not approach a single, finite value.