Solving Limits (3 Ways)

Why it matters

• Limits often give indeterminate forms like 0/0.
• To solve them, we need smart techniques instead of plugging values directly.
• These three methods — Substitution, Factoring, Conjugate — are the most common and easiest to learn.

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1. Substitution (Direct Plug-In)

• The simplest method: just put the value of x into the function.
• If the result is a finite number (not 0/0, k/0, or ∞ - ∞), you are done.

✅ Example:

lim {x tends to 2} (x^2 + 3) = 2^2 + 3 = 7

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2. Factoring

• Use this when substitution gives 0/0.
• Factor numerator and denominator, cancel common terms, then substitute again.

✅ Example:

lim{x tends to 1} (x^2 - 1)/(x - 1)

Factor:

= lim{x tends to 1} (x-1)(x+1)/(x-1)

Cancel (x-1):

= lim{x tends to 1} (x+1) = 2

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3. Conjugate (Multiply by Opposite Sign)

• Use this when roots (sqrt) are involved and substitution gives 0/0.
• Multiply top and bottom by the conjugate (same expression but with opposite sign).

✅ Example:

lim{x tends to 0} (sqrt(x+4) - 2)/x

Multiply by conjugate:

(sqrt(x+4) - 2)/x * (sqrt(x+4) + 2)/(sqrt(x+4) + 2)

Simplify numerator:

= ((x+4) - 4)/x(sqrt(x+4)+2) = x/x(sqrt(x+4)+2)

Cancel x:

= 1/(sqrt(x+4)+2)

Now substitute x=0:

= 1/2+2 = 1/4

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✅ In short:

• Substitution → Try plugging in directly.
• Factoring → If you get 0/0, factor and cancel.
• Conjugate → If roots are involved, multiply by conjugate.