Limits

Calculus 1.1 and 1.2

DerivativesProblem: Find the area of this picture.

04/19/23 – LO: Limits - Determine if they exist.#102 p54 5, 8, 12, 15-24, 26, 28

Intro to CalculusDerivativesProblem: Find the area of this picture.

Intro to CalculusDerivativesProblem: Find the area of this picture.

The Tangent Line Problem

The Tangent Line Problem

x∆x = distance from x

x + ∆x 1∆x = 4

5

yf(x)

y + ∆yf(x+∆x)

f(1)

f(5)

∆y=2

=10

∆y = 8

The Tangent Line Problem

f(1 + 4)

The Tangent Line Problem

xx

Develop a habit to try 3 approaches to problem solving.

1. Numerical ApproachConstruct a table of values.

2. Graphical ApproachDraw a graph.

3. Analytical ApproachUse algebra or calculus.

3 Limits that FAIL

1. Behavior that is different from Right and Left

As x creeps to the limit, the function goes towards different values from the right and the left.

3 Limits that FAIL

2. Unbounded Behavior

Function goes to positive or negative infinite.

3 Limits that FAIL

3. Oscillating Behavior

Trig functions with x in the denominator.

Limits that FAIL

Common types of behavior where the limit does not exist.

• f(x) approaches different values from left and right.

• f(x) increase or decrease without bound as x approaches c.

• f(x) oscillates between 2 different values as x approaches c.

0.204 0.2004 0.2 0.1999 0.1996 0.196

= 0.2

1. Put function into Graph mode (MENU 5).2. Verify appropriate window (V-Window)3. Draw graph - Analyze graph at limit4. Use TRACE to complete table

1. Put function into Graph mode (MENU 5).2. Verify RAD mode (Shift - SETUP)3. V-Window to TRIG4. Draw graph - Analyze graph at limit5. Use TRACE to complete table

0.99833 0.99998 0.99999 0.99999 0.99998 0.99833

= 1

x

f(x)

10.9990.990.9 1.001 1.01 1.1

IND F0.66730.67330.7340 0.6660 0.6600 0.6015

= 0.667 = 2/3

= 1 = 4

= 2 = 4

Limit does notexist. The functionapproaches 1 fromthe right and -1 fromthe left as xapproaches 2.

Limit does notexist. The functiondecreases from theleft without bound and increases fromthe right without bound as x approaches 5.

Yes. f(1) = 2

(b) No, the limit does not exist. As the x approaches 1, the function approaches 1 from the right but approaches 3.5 from the left.(c) No, the value does not exist. The function is undefined when x = 4.(d) Yes. f(4) = 2.

At x = -3, the function does not exist.

At x = 2, the function DOES exist.