Evaluate a Limit by Direct Substitution

Direct substitution idea

If the function is well-behaved at the target value, you can find the limit by plugging that value into the expression. This works when the function is continuous at that point, so the left and right values approach the same number.

Method

1. Identify the input value the variable approaches.
2. Substitute it into the expression exactly as written.
3. Simplify carefully using order of operations.
4. Reduce the result to its simplest form.

What to watch for

If direct substitution gives an undefined form such as division by zero, then the limit may need another method. But for direct substitution exercises, the expression should usually be defined after plugging in the value.

Quick check

After simplifying, re-evaluate mentally: does the answer match what you get by direct replacement? If the expression is a polynomial, a rational expression with a nonzero denominator, or another continuous expression at that point, the substitution result is the limit.

Example structure

For a limit like (\lim_{x\to a} f(x)), compute (f(a)) and simplify. The final simplified value is the limit.