Substitute the value directly into the function
Simplify algebraically if direct substitution gives an indeterminate form
Factor and cancel common terms
Rationalize the numerator or denominator if needed
Combine fractions to get a common denominator
Use trigonometric identities when trigonometric expressions are involved
Apply standard limit laws
Use special limits such as (lim_{x to 0} frac{sin x}{x} = 1)
Evaluate one-sided limits when the function behaves differently from the left and right
Check for discontinuities, holes, and asymptotes
Use graphing or tables to estimate the limit if needed
Apply L’Hôpital’s Rule for indeterminate forms when allowed
Use the squeeze theorem when the function is trapped between two others with the same limit
Rewrite expressions to match known limit forms
Consider limits at infinity by comparing dominant terms
Verify whether the left-hand and right-hand limits are equal
State that the limit does not exist if the one-sided limits are different or the function oscillates