Look at the y-value the graph approaches as x approaches a certain value. The limit is what y is heading toward, not necessarily the value at that point.

Analyze the trend of y-values as x-values get closer to a specific point. Look for patterns and the value y is approaching.

Use limit laws to break down complex functions into simpler ones. Limits work well with addition, subtraction, multiplication, and division. Consider composite functions.

Manipulate the function to eliminate indeterminate forms (like 0/0). Use conjugates, L'Hôpital's rule, or simplifying rational functions.

When you are given a graph. Scan the graph to see what y-value the function approaches as x approaches a certain value.

When you are given a table of x and y values. Look for patterns and see where the y-values are heading as the x-values approach a specific point.

When you see basic limit theorems (sums, differences, products, quotients) or composite functions. Break down the problem into simpler parts.

When you have complex functions that need simplification. Think conjugates, L'Hôpital's Rule, or simplifying rational functions.

If a function is bounded above and below by two other functions that have the same limit at a certain point, then the function in the middle must also have the same limit at that point.

L'Hopital's Rule can only be applied to indeterminate forms (0/0 or ∞/∞). Applying it to other forms will lead to incorrect results.

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.

$\lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x)$

$\lim_{x \to c} [f(x) - g(x)] = \lim_{x \to c} f(x) - \lim_{x \to c} g(x)$

$\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)$

$\lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)}$, provided $\lim_{x \to c} g(x) \neq 0$

If $\lim_{x \to c} f(x) = 0$ and $\lim_{x \to c} g(x) = 0$, then $\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$

$\lim_{x \to c} [k \cdot f(x)] = k \cdot \lim_{x \to c} f(x)$

$\lim_{x \to c} [f(x)]^n = [\lim_{x \to c} f(x)]^n$

$\lim_{x \to c} k = k$

$\lim_{x \to c} x = c$

If $f(x) \leq g(x) \leq h(x)$ for all x near c, and $\lim_{x \to c} f(x) = L = \lim_{x \to c} h(x)$, then $\lim_{x \to c} g(x) = L$.