Understanding Limits: A Focus on Infinity

Horizontal Asymptotes and Limits at Infinity

When x grows extremely large (positively or negatively), some functions approach a specific value—this creates a horizontal asymptote. For example, as x approaches infinity in the function f$x$ = 1/x, the function approaches 0.

A horizontal asymptote exists when either lim x→∞ f$x$ = 0 or lim x→-∞ f$x$ = 0. These asymptotes show the end behavior of a function as x gets extremely large in either direction.

For rational functions (fractions with polynomials), finding horizontal asymptotes follows three simple rules:

• If the denominator's degree > numerator's degree: horizontal asymptote at y = 0
• If the numerator's degree > denominator's degree: creates a slant asymptote
• If degrees are equal: horizontal asymptote at y = leading coefficient of numerator/leading coefficient of denominator

💡 Think of horizontal asymptotes as the "finish line" your function approaches but never quite reaches as x races toward infinity!

Finding Limits at Infinity

To find limits as x approaches infinity for rational functions, divide every term by the highest power in the denominator. This clever technique simplifies the problem by making terms with x in the denominator approach zero.

For example, to find lim x→∞ $5x² - 3x + 1$/$2x² + 4x - 7$, divide everything by x²:

• This gives: lim x→∞ $5 - 3/x + 1/x²$/$2 + 4/x - 7/x²$
• As x→∞, any term with x in the denominator approaches 0
• So we get: 5/2 = 2.5

When evaluating these limits, remember that different degrees in numerator and denominator lead to different results:

• Higher degree in numerator: limit = ±∞
• Higher degree in denominator: limit = 0
• Equal degrees: limit = ratio of leading coefficients

🔑 The trick to solving these problems is focusing on which terms survive as x gets extremely large. Terms with higher powers of x in the denominator vanish faster!

Vertical Asymptotes and One-Sided Limits

Vertical asymptotes occur at points where the denominator equals zero (but the numerator doesn't). These are points where the function shoots off to infinity in at least one direction.

When analyzing a rational function near a potential vertical asymptote at x = a, we need to check the one-sided limits:

• lim x→a⁺ f$x$ examines what happens approaching from the right
• lim x→a⁻ f$x$ examines what happens approaching from the left

For a function like f$x$ = 3/x, we see that:

• lim x→0⁺ 3/x = ∞ (approaching zero from positive numbers)
• lim x→0⁻ 3/x = -∞ (approaching zero from negative numbers)
• Therefore, the overall limit does not exist (DNE)

🚨 Don't confuse vertical asymptotes with holes! If both numerator and denominator equal zero at the same point (a "0/0" situation), you get a hole rather than a vertical asymptote.

Evaluating Limits at Vertical Asymptotes

When finding limits at potential vertical asymptotes, you must check both the left and right sides separately to determine if the limit exists. If the one-sided limits don't match, the overall limit doesn't exist.

For example, in lim x→-3 $x²+1$/$3+x$:

• From the left: lim x→-3⁻ $x²+1$/$3+x$ = (positive)/(negative) = -∞
• From the right: lim x→-3⁺ $x²+1$/$3+x$ = (positive)/(positive) = ∞
• Since these one-sided limits differ, the overall limit doesn't exist

The sign of the result depends on the signs of the numerator and denominator:

• (positive)/(positive) = ∞
• (positive)/(negative) = -∞
• (negative)/(positive) = -∞
• (negative)/(negative) = ∞

Remember that for limits involving squared terms in the denominator like lim x→1 2/$x-1$², both one-sided limits give ∞, so the overall limit equals infinity.

📝 A good strategy: First identify where the denominator equals zero, then check what happens as you approach from both sides by analyzing the signs of the numerator and denominator.