Limits, Continuity, Basic Differentiability, and Computing Derivatives By: Sameer, Snigdha, Aditya

Limits Recall that… A limit is when a function gets super close to a number from both sides of x, but the function never reaches that number… It’s predicting a number between two neighboring points. Let’s think about how we can solve these

Solving Limits - Ex1: Make a table -0.1 -.01 -.001 0.001 0.01 0.1 Y Plug in on your calculator!

Solving Limits - Ex1: Make a table (cont.) -0.1 -.01 -.001 0.001 0.01 0.1 Y 0.998 0.999 1 Therefore, the answer is 1 because as x approaches 0 from both sides, the value of the limit gets closer and closer to 1.

Solving Limits - Ex2: Graph As x approaches 0 from both sides, what does f (x) get arbitrarily close to? lim f(x) = 2 x->0

Solving Limits - Ex3: Direct Substitution When a function is given, your first tactic to solving a limit should be plugging in the value the limit approaches. Plug in x = 5!

Solving Limits - Ex4: Factoring/Simplifying If direct substitution yields an indeterminate answer (0/0 or infinity/infinity), try factoring!

Solving Limits - Ex4: Factoring/Simplifying Let’s try factoring.... And now let’s simplify... Now when we use direct substitution, we get 8/2, which is 4

Solving Limits - Ex5: Rationalizing the Numerator/Denominator When you can’t use direct substitution or factoring, check for a radical and try rationalizing the numerator or denominator

Solving Limits - Ex5: Rationalizing the Numerator/Denominator Multiply either the numerator or denominator by a radical that will get rid of the radical sign and use a different sign for the second term…

Solving Limits - Ex5: Rationalizing the Numerator/Denominator Then, simplify:

Solving Limits - Ex5: Rationalizing the Numerator/Denominator Finally, use direct substitution:

Limits - Does Not Exist (DNE) This limit does not exist because on the limit from the left of x approaches 1 while on the right side of x it approaches 2.

Limits - Vertical Asymptotes If f(x) approaches infinity or negative infinity as x approaches c from the right or left, then there is a vertical asymptote at x = c lim f(x) = infinity; therefore VA x->0

Limits - Vertical Asymptotes (cont.) A constant divided by a small number yields infinity: This indicates a potential vertical asymptote

Limits - Horizontal Asymptote The line y = L is a horizontal tangent if limit as x approaches infinity for f(x) = L

Limits - Horizontal Asymptotes (cont.) Use coefficient rule: if highest power in the numerator and denominator match, then divide the coefficients for the limit/horizontal asymptote =3/2 If the power is smaller in numerator or the numerator has a slower function then the limit goes to 0.

Continuity Consider this hole in the road...is the road continuous?

Continuity - Definition Most of you probably agreed that the road is not continuous...you wouldn’t want to drive over that! But...what would make the road continuous? Well, the road would have to: -Fill the hole (discontinuity filled by f(c)) -There would have to be road on either side of the road (limit must exist) -The road on either side and the filled in hole would have to connect (limit must equal f(c)

Continuity - Definition (cont.) Similarly in calculus these rules also apply! The limit as x approaches c for f(x) must exist f(c) must exist and have a value The limit as x approaches c for f(x) must equal f(c) Now, justify if each of the points are continuous or not.

Continuity - Justification Continuous.The limit as x approaches 0 is 1 and f(0) is also 1 Discontinuous. The limit as x approaches -2 does not exist Discontinuous. The limit as x approaches 3 is 0, but f(3) is -1. Therefore there is a discontinuity at x = 3

Continuity - Practice Problem Hmm...let’s use the definition of continuity to solve this problem

Continuity - Practice Problem (cont.) First, let’s look at x=1. f(1) = 2 so the limit as x approaches 1 must also equal 2. Use direct substitution!

Continuity - Practice Problem (cont.) Plugging in x=1 gives us a + b = 2 Now let’s look at x = 3. f(3) gives -2 so the limit as x approaches 3 must also be -2. Again, use direct substitution… Plugging in x=3 gives us 3a + b = -2

Continuity - Practice Problem (cont.) Now, setup a system of equations: a + b = 2 3a + b = -2 Solving for a gives - 2a = 4 a = -2. Plugging back in, we get -2 + b = 2, and b = 4

Basic Differentiability A function is differentiable if its derivative exists at every point on its domain. A derivative is essentially the slope of a function and is denoted as f’(x) Differentiability implies Continuity However, continuity does not mean a function is 100% differentiable

Basic Differentiability - Non-differentiable functions Corner Cusp

Basic Differentiability - Non-differentiable functions (cont.) Vertical Tangent Jump Discontinuity

Computing a Derivative Formal Definition of a Derivative This simply means the derivative of cos(x) at x= pi

Computing a Derivative - Polynomials Here’s the formula for computing a derivative* *Note: there are special ways to derive trig, logarithmic, e^x, and other special case functions

Computing a Derivative - Polynomials (cont.)

Computing a Derivative - Rational Functions Quotient Rule: When computing functions with quotients, you must use quotient rule.

Computing a Derivative - Rational Functions (cont.) Use quotient rule!

Computing a Derivative - Rational Functions (cont.) Derivative of f(x)!

Computing a Derivative - Trigonometric Functions Memorize these Trig Derivatives!

Computing a Derivative - Trigonometric Functions (cont.) Quick introduction to chain rule: a derivative rule in which nested functions must be included in the derivative.

Computing a Derivative - Trigonometric Functions (cont.) h(x) = sin(2x)...Find the derivative f(x) = sin(x) g(x) = 2x so: h’(x) = cos(2x)*2 or 2cos(2x)

Computing a Derivative - Exponential/Logarithmic Functions You should know these derivatives:

Computing a Derivative - Exponential/Logarithmic Functions (cont.) Finally, product rule:`

Computing a Derivative - Exponential/Logarithmic Functions (cont.) Let’s take the derivative of this! Using chain rule, we get:

That’s All Folks… FIN.