Compare Leading Linear Terms at Large Inputs

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Find lim as x→∞ of (2x+1)/(3x−2). Candidate answers: (A) -1 (B) 0 (C) 2/3 (D) 3/2.
• Independent check inputs: Find lim as x→∞ of (3x+1)/(4x−2). Candidate answers: (A) -1 (B) 0 (C) 3/4 (D) 4/3.

Learn the idea

A limit describes nearby approach behavior rather than necessarily a function’s value at the point. One-sided agreement, removable holes and continuity connect algebraic simplification with precise statements about a function’s local behavior. Dividing numerator and denominator by x reveals reciprocal terms that approach zero as x grows without bound. With a nonzero denominator leading coefficient, the remaining coefficient ratio determines this limit.

Here is the method on the supplied teaching case. Divide numerator and denominator by x: (2+1/x)/(3−2/x). The reciprocal terms approach zero, leaving 2/3. The denominator approaches nonzero 3.

Infinity is an approach direction rather than a number to substitute. The equal-degree leading-coefficient shortcut here does not apply unchanged when numerator and denominator degrees differ or other functions are involved.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Take it further

Changed-input task: Find lim as x→∞ of (9x+1)/(10x−2). Candidate answers: (A) -1 (B) 0 (C) 10/9 (D) 9/10. Show the mathematical reason, not only the selected answer. Expected result: 9/10. Divide numerator and denominator by x: (9+1/x)/(10−2/x). The reciprocal terms approach zero, leaving 9/10. The denominator approaches nonzero 10.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.