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Quizzes > High School Quizzes > Mathematics

Master Limits: Practice Quiz for Success

Sharpen skills with interactive questions and clear explanations

Difficulty: Moderate
Grade: Grade 12
Study OutcomesCheat Sheet
Colorful paper art promoting a calculus trivia quiz for high school students.

Evaluate the limit: lim(x→3) (2x + 1).
9
6
8
7
By directly substituting x = 3 into the function 2x + 1, we compute 2(3) + 1 = 7. Since the function is linear and continuous, the limit is 7.
Find the limit: lim(x→-2) 5.
-2
5
-5
0
A constant function always returns the same value regardless of x. Therefore, the limit as x approaches any number, including -2, is 5.
Determine the limit: lim(x→0) (x²).
0
 
1
-1
Substituting x = 0 into x² gives 0² = 0. Since x² is continuous everywhere, the limit is simply 0.
Compute the limit: lim(x→4) (x + 2).
4
2
8
6
By substituting x = 4 into the linear function x + 2, we obtain 4 + 2 = 6. Continuity of linear functions guarantees this result is the limit.
What is the limit of the constant function f(x) = 3 as x approaches 1?
3
0
 
1
A constant function always has the same output, independent of the input value. Thus, the limit as x approaches any number is the constant value 3.
Evaluate the limit: lim(x→0) (sin(x)/x).
 
1
0
infinity
The standard trigonometric limit states that sin(x)/x approaches 1 as x approaches 0. This is one of the most fundamental limits in calculus.
Find the limit: lim(x→2) ((x² - 4)/(x - 2)).
0
2
 
4
Factor the numerator as (x - 2)(x + 2) and cancel the common factor with the denominator. Substituting x = 2 in the simplified expression x + 2 yields 4.
Determine the limit: lim(x→0) ((1 - cos(x))/(x²)).
1/2
 
0
1
Using the Taylor series expansion for cos(x) where cos(x) ≈ 1 - x²/2 for small x, the numerator approximates x²/2. Dividing by x² results in a limit of 1/2.
Evaluate the limit: lim(x→0) ((eˣ - 1)/x).
0
 
e
1
The limit represents the derivative of eˣ at x = 0, which is e❰ = 1. Series expansion of eˣ further confirms that the limit is 1.
Compute the limit: lim(x→∞) ((2x² + 3)/(x² - 1)).
3
infinity
0
2
For large values of x, the highest power terms dominate the behavior of the function. Dividing both numerator and denominator by x² gives the ratio 2/1, so the limit is 2.
Determine the limit: lim(x→2) ((x³ - 8)/(x - 2)).
6
4
12
8
Factor the numerator as (x - 2)(x² + 2x + 4) and cancel the common factor with the denominator. Plugging x = 2 into the remaining quadratic produces 12.
Evaluate the limit: lim(x→0) ((tan(x))/(x)).
infinity
1
0
 
For small angles, tan(x) is approximately equal to x. Therefore, the ratio tan(x)/x approaches 1 as x tends to 0.
Find the limit: lim(x→0) (x · sin(1/x)).
1
 
0
does not exist
Even though sin(1/x) oscillates between -1 and 1, the factor x approaches 0. By the squeeze theorem, the entire product is forced to 0.
Determine the limit: lim(x→(π/2)❻) (tan(x)).
0
1
-infinity
infinity
As x approaches π/2 from the left, the cosine function approaches 0 while sine remains positive, causing tan(x) = sin(x)/cos(x) to increase without bound. Thus, the limit is infinity.
Evaluate the limit: lim(x→0) ((sin(2x))/(x)).
4
0
1
2
Rewrite sin(2x)/x as 2 · (sin(2x)/(2x)), and then apply the standard limit which equals 1. Multiplying by 2 gives a limit of 2.
Compute the limit: lim(x→0) ((√(1 + x) - √(1 - x))/x).
0
1
2
1/2
Multiply the numerator and denominator by the conjugate √(1 + x) + √(1 - x) to simplify the expression. After cancellation and substitution, the limit evaluates to 1.
Evaluate the limit: lim(x→0) ((1 - cos(3x))/(2x²)).
9/2
1/2
3/2
9/4
Using the approximation cos(3x) ≈ 1 - (9x²/2) for small x, the numerator approximates 9x²/2. Dividing this by 2x² leads to a limit of 9/4.
Determine the limit: lim(x→0) ((tan(2x))/(sin(3x))).
3/2
1
2/3
2
For small x, tan(2x) approximates 2x and sin(3x) approximates 3x. Dividing gives (2x)/(3x) = 2/3.
Evaluate the limit: lim(x→∞) (3x - √(9x² + x)).
infinity
1/6
-1/6
0
Factor out x from the square root to write √(9x² + x) as x√(9 + 1/x). Expanding for large x, the expression simplifies and the limit evaluates to -1/6.
Compute the limit: lim(x→0) ((sin x - x cos x)/x³).
1/2
0
1/3
1
Use Taylor series expansions for sin x and cos x: sin x ≈ x - x³/6 and cos x ≈ 1 - x²/2. Simplifying the expression and dividing by x³ yields a limit of 1/3.
0
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Study Outcomes

  1. Understand and articulate the formal definition of limits.
  2. Analyze limit behavior of functions using various limit techniques.
  3. Apply limit laws to evaluate complex algebraic, rational, and trigonometric expressions.
  4. Evaluate one-sided limits and identify points of discontinuity.
  5. Use the squeeze theorem to resolve indeterminate forms.
  6. Interpret limit results to assess function trends and asymptotic behavior.

Limits Quiz: Practice & Review Cheat Sheet

  1. Understand the Definition of a Limit - In calculus, a limit is like a GPS guiding you to where a function wants to go as x nears a target point. Imagine a ninja sneaking closer to a treasure chest - that's a limit in action! Mastering this concept is your first step to acing calculus. Learn more about limits
  2. Master Limit Laws - The sum, difference, product, and quotient rules are your calculus toolbox for breaking down complex limits into bite-sized pieces. It's like following a recipe: mix, match, and simplify for faster results. Get the limit laws cheat sheet
  3. Practice Direct Substitution - Often you can just plug in the value x is approaching directly into the function. If f(x)=x² then lim(x→3) f(x)=9 - no magic needed! Keep substituting to build confidence. Try substitution problems
  4. Handle Indeterminate Forms - Expressions like 0/0 are calculus puzzles that need factoring, conjugates, or L'Hôpital's Rule to crack. Picture a locked door - you need the correct key or technique to open it! Unlock indeterminate forms
  5. Understand One-Sided Limits - Sometimes you can only approach a point from the left (−) or the right (+), like sneaking into a building through a single door. These limits help you detect jumps and holes in functions. Explore one-sided limits
  6. Explore Limits at Infinity - Ever wondered what happens as x zooms off to +∞ or −∞? Limits at infinity reveal horizontal asymptotes and end behaviors, helping you sketch the big picture of a graph. Study limits at infinity
  7. Apply the Squeeze Theorem - When a function is sandwiched between two others with the same limit, you can "squeeze" out the answer. It's like solving a mystery by surrounding the suspect from both sides! Learn the Squeeze Theorem
  8. Recognize Common Limit Results - Memorize classic limits such as lim(x→0)(sin x)/x = 1 and lim(x→0)(1 − cos x)/x = 0. These are your calculus speed hacks for lightning-fast solutions. Review standard limits
  9. Practice with Piecewise Functions - Functions defined by different rules on different intervals can hide surprises at their boundaries. Check continuity at transition points like you're hunting for Easter eggs. Compute piecewise limits
  10. Utilize Graphical Analysis - Sometimes a picture is worth a thousand limits. Sketching or using graphing tools helps you visualize asymptotes, holes, and behaviors you might miss algebraically. Visualize limits with graphs
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