Analyze limits from each side and describe end behavior using limits at infinity.
The left-hand limit lim_{x→a⁻} f(x) considers x approaching a from values less than a. The right-hand limit lim_{x→a⁺} f(x) considers x approaching from above. The two-sided limit exists only when both one-sided limits exist and are equal.
x→alimf(x)=L⟺x→a−limf(x)=L and x→a+limf(x)=L
x→∞limx1=0x→∞lim2x2−13x2+5x=23(ratio of leading coefficients)
Limits at Infinity for Rational Functions
1Divide numerator and denominator by the highest power of x in the denominator.
2Each term with x in the denominator goes to 0 as x → ∞.
3The limit equals the ratio of the leading coefficients (if degrees are equal).
4The limit is 0 if numerator degree < denominator degree.
5The limit is ±∞ if numerator degree > denominator degree.
x→∞lim3x3+x25x3−2x=x→∞lim3+1/x5−2/x2=35
📈Piecewise Limit
f(x) = {x+1 if x < 2, x²−1 if x ≥ 2}. Left limit as x→2⁻: 2+1 = 3. Right limit as x→2⁺: 4−1 = 3. Both equal 3, so lim_{x→2} f(x) = 3.
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Remember This!
For limits at infinity involving square roots, multiply by the conjugate or factor out the dominant power. For example, lim √(x²+1)/x = lim √(1 + 1/x²) = 1 as x → +∞.