Limits

Limits at Infinity

Find the limit of a rational function as x approaches infinity by dividing by the highest power.

limit-at-infinity Divide the numerator and denominator by the highest power of x that appears. Terms with x in the denominator vanish as x → ∞, leaving only the leading coefficients.

Example

Use highest-power terms to find an end-behavior limit exactly.

highlighted = computed this step

Step 1 — Set up

Set up the limit as x approaches infinity.

lim⁡x→∞3x2−x2x2+5\lim_{x\to\infty} \frac{3x^{2}-x}{2x^{2}+5}

Step 2 — Divide by highest power

Divide every term by x squared.

3−1x2+5x2\hlmath{\frac{3-\frac{1}{x}}{2+\frac{5}{x^{2}}}}

Step 3 — Let small terms vanish

As x grows, the smaller fraction terms go to 0.

1x→05x2→0\frac{ 1 }{x} \to \hl{0} \quad\frac{ 5 }{x^{ 2 }}\to \hl{0}

Step 4 — Result

Keep the leading-coefficient ratio 3 over 2.

lim⁡=32\lim= \hlmath{\frac{3}{2}}