2.0 REVIEW: LIMIT OF A FUNCTION
Idea (Polygon to Circle Analogy)
As the number of sides of a polygon increases, it approximates a circle. This helps visualize how a function value approaches a particular number as x approaches a point.
Meaning of Limit
The limit of a function at x = a is the value that f(x) approaches as x comes very close to a from both sides.
Example 1: f(x) = x² at x = 2
Stepwise Solution:
1.Check values slightly less than 2 (left-hand limit):
x=1.9 → f(x)=3.61
x=1.99 → f(x)=3.9601
x=1.999 → f(x)=3.996001
2.Check values slightly greater than 2 (right-hand limit):
x=2.0001 → f(x)=4.00040001
x=2.001 → f(x)=4.004001
x=2.01 → f(x)=4.0401
3.Since left-hand and right-hand limits approach 4 → Limit exists → Limit = 4
Example 2: f(x) = (x² - 1)/(x - 1) at x = 1
Stepwise Solution:
1.Function undefined at x = 1
2.Check values near 1:
x = 0.99 → f(x) ≈ 2
x = 1.01 → f(x) ≈ 2
3.Both sides approach 2 → Limit exists
4.Since f(1) undefined → removable discontinuity
Example 3: f(x) = 3x + 1 at x = 2
Stepwise Solution:
1.Substitute x = 2 → f(2) = 3×2 + 1 = 7
2.Left and right values = 7 → Limit exists
3.f(2) defined → Continuous at x = 2
2.1 CONTINUITY IN DIFFERENT SETS OF NUMBERS
Natural Numbers → Discontinuous (gaps between numbers)
Integers → Discontinuous
Rational Numbers → Not continuous (irrationals exist between any two rationals)
Real Numbers → Continuous (no gaps; infinite numbers between any two real numbers)
Real-Life Examples of Continuity: Flow of river, temperature change, growth of plant height
Real-Life Examples of Discontinuity: Attendance, number of cars passing a signal, jumping frog
2.2 CONTINUITY AND DISCONTINUITY IN GRAPHS
Continuous Graph → Drawn without lifting pen, no breaks or jumps
Discontinuous Graph → Graph has break, jump, gap, or hole
Example 4: y = x + 2 → Graph smooth → Continuous
Example 5: Step Function → Graph jumps → Discontinuous at jump points
2.3 SYMBOLIC REPRESENTATION OF CONTINUITY
A function f(x) is continuous at x = a if:
1.f(a) is defined
2.Left-hand limit = Right-hand limit
3.Limit = f(a)
Example 6: f(x) = (x² - 9)/(x - 3)
Stepwise Solution:
1.f(x) undefined at x=3
2.Simplify: f(x) = x + 3 for x ≠ 3
3.Check values near 3:
x=2.99 → f(x) ≈ 5.99
x=3.01 → f(x) ≈ 6.01
4.Limit exists → 6, function undefined → Removable discontinuity
Example 7: Quadratic f(x) = x² + 4x + 1 → No breaks → Continuous for all real x
Example 8: Piecewise Function
f(x) = x + 1 for x < 2
f(x) = 3x - 1 for x ≥ 2
Stepwise Solution:
1.Check left value at x=2 → f(2⁻) = 2 + 1 = 3
2.Check right value at x=2 → f(2⁺) = 3×2 - 1 = 5
3.Left ≠ Right → Function discontinuous at x=2
2.4 TYPES OF DISCONTINUITY
Removable Discontinuity → Limit exists but f(a) undefined or different → Example: f(x)=(x²-1)/(x-1) at x=1
Jump Discontinuity → Left-hand limit ≠ Right-hand limit → Example: Step function
Infinite Discontinuity → Function approaches infinity at a point → Example: f(x)=1/x at x=0
So function is discontinuous at x = 1.
2.5 IMPORTANT PRACTICE QUESTIONS WITH STEPWISE SOLUTIONS
Q1: Find limit of f(x) = 2x + 5 at x = 3
Solution:
1.Substitute x = 3 → f(3) = 2×3 + 5 = 11
2.Left and right limits = 11 → Limit exists → Limit = 11
Q2: Check continuity of f(x) = x² at x = 1
Solution:
1.f(1) = 1² = 1
2.Left and right-hand limits = 1 → Limit exists
3.f(1) defined → Function continuous at x = 1
Q3: Find limit of (x² - 4)/(x - 2) at x = 2
Solution:
1.Factor numerator: x² - 4 = (x - 2)(x + 2)
2.Simplify: f(x) = (x + 2) for x ≠ 2
3.Limit as x → 2 → f(x) = 4
4.f(2) undefined → Removable discontinuity
Q4: Give two real-life examples of continuity and discontinuity
Solution:
Continuous → Temperature change, river flow
Discontinuous → Attendance, step function of traffic signal
Q5: Check continuity of piecewise function at x = 2
f(x) = x + 1 (x<2), f(x) = 3x -1 (x≥2)
Stepwise:
1.Left limit f(2⁻) = 2 + 1 = 3
2.Right limit f(2⁺) = 3×2 -1 = 5
3.Left ≠ Right → Discontinuous at x = 2
Q6: Determine type of discontinuity for f(x) = (x² - 1)/(x - 1) at x=1
Solution:
Simplify: f(x) = x + 1 for x ≠ 1
Limit as x → 1 → 2, f(1) undefined → Removable discontinuity
Q7: Check whether f(x) = 1/x is continuous at x=0
Solution:
1.f(0) undefined
2.Left-hand limit → -∞, Right-hand limit → +∞
3.Limit does not exist → Infinite discontinuity
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