Infinite Limits Sketch Worksheet
Calculus • Curve Sketching & Limits
Infinite Limits at Asymptotes
Visual Guide Target: \( x \to -1 \)
f(x)
Target Conditions to Sketch
1. \( \lim_{x \to -1^-} f(x) = -\infty \) 2. \( \lim_{x \to -1^+} f(x) = -\infty \)
Vertical Asymptote
Line: \( x = -1 \)
Graphical Solution & Behavior Along Asymptote
Asymptote \( x = -1 \) Curve \( f(x) \)
x y -4 -3 -2 -1 1 2 3 V.A. x = -1 From the Left (\( x \to -1^- \)) Values plunge to \( -\infty \) From the Right (\( x \to -1^+ \)) Values plunge to \( -\infty \)
Step-by-Step Limit Translation
1. Left-Hand Limit: \( \lim_{x \to -1^-} f(x) = -\infty \)
As \( x \) approaches \( -1 \) from values less than \( -1 \) (e.g., \(-1.1, -1.01\)), the graph must drop downward without bound toward negative infinity.
2. Right-Hand Limit: \( \lim_{x \to -1^+} f(x) = -\infty \)
As \( x \) approaches \( -1 \) from values greater than \( -1 \) (e.g., \(-0.9, -0.99\)), the graph similarly drops downward without bound toward negative infinity.
Algebraic Model & Two-Sided Behavior
Standard Equation Prototype
\( f(x) = -\frac{1}{(x + 1)^2} \)
Since \( (x+1)^2 > 0 \) for all \( x \neq -1 \), the denominator is always positive and approaches \( 0^+ \). The negative numerator forces \( f(x) \to -\infty \) on both sides.
Two-Sided Limit Conclusion: Both sides agree in direction:
\( \lim_{x \to -1} f(x) = -\infty \)
Pro-Tip for Graphing Limits
Always draw the vertical asymptote as a dashed vertical reference line first before sketching the approaching curves!
Page 1 • Concept Reference
Active Practice Lab
Infinite Limits Sketching Lab
Name ____________________
Date / Period ____________________
Directions: For each problem, sketch a continuous-except-at-asymptotes function that strictly satisfies the stated limit criteria. Clearly draw and label all vertical asymptotes with dashed lines.
1
Opposite Infinite Limits at \( x = 2 \)
\( \lim_{x \to 2^-} g(x) = +\infty \) \( \lim_{x \to 2^+} g(x) = -\infty \)
x y -2 2 4
Asymptote Equation: x = ________
Two-sided limit \( \lim_{x \to 2} g(x) \): [ ] \( +\infty \) [ ] \( -\infty \) [ ] Does Not Exist
Sample Prototype: g(x) = \( -\frac{1}{x - 2} \)
2
Mixed Behavior: Infinite & Finite at \( x = 0 \)
\( \lim_{x \to 0^-} h(x) = -\infty \) \( \lim_{x \to 0^+} h(x) = 2 \) \( h(0) = -1 \)
x y 2 1 -1 -2
Point at \( x = 0 \): Does \( h(0) \) require a solid dot, open circle, or asymptote? Answer: ______________
Right-Side Feature: At \( (0, 2) \), sketch: [ ] Open Hole [ ] Solid Dot
Sketching Checklist: • Dashed line for asymptote • Correct left/right plunge direction • Proper open/solid circles
Self-Score: ____ / 10 pts
Calculus Curriculum Series • Single-Variable Limits Page 2 of 2 • Student Sketching Sheet
