2.2 Formal Definition of a Limit1. What is the formal definition of a limit?2. T or F: when we are talking about a limit around some point c, we use an open punctured interval centered around c that does not include the point c itself.3. Fill in the blanks: (note, drawing a picture may help)a. For lim x c+ f(x)= infinity (infinite limits)for all _________, there is some _________ such that if _________, then _____________.b. For lim x infinity f(x)= Lfor all _________, there is some _________ such that if _________, then _____________.4. Describe what the uniqueness of a limit means5. What is the formal definition of an infinite limit at infinity?6. Write the limit below as a formal statement involving delta or N, and epsilon or Mlim x 1 (x2-3) = -27. Find the largest value of delta for which [f(x)-L] < epsilon whenever 0<[x-c]<delta(note: [] signs symbolize absolute value signs)f(x)= x 2 – 2x – 3 L=-4, c=-1, epsilon=1 x+1Make sure you know the definition of a limit using delta and epsilon, and understand where it came from!!!!!2.3 Delta Epsilon ProofsFollow this example’s guidelines to complete the later delta-epsilon’s proofs Ex: lim x 2 (3x+1)=7PROVE: Show for all  >0, there exists some >0, such that if 0<[x-2]< , then [3x+1-7]<  .PROOF: Given  >0 Choose =  /3 If 0<[x-2]< , then [3x+1-7] = [3x-6] = [3(x-2)] = 3[x-2] < 3 3( /3) = 1. Prove that 2. Prove that 3. Prove that2.4 Limit Rules1. Write the following definitions: (it will be helpful to know the proofs of these as well)a. Limit of a constantb. The limit of the identity functionc. The limit of a linear functiond. The constant multiple rules for limitse. The sum rule for limits and the difference rule for limitsf. The limit of a power function with a positive integer powerg. The multiplication rule and the division rule for limits:- In order for the multiplication rule and the division rule for limits to apply, what must be true?2. Prove part e in number 1 by induction:3. Evaluate the following limits, justifying your steps along the waya. lim x 0 (x2-1) b. lim x -3 2/(3x+1)2.5 Calculating Limits1. What are the two assumptions that must be met in order for one to simply “plug in” the value “c” within the framework of the limit function? -- (Hint: see algorithm 2.1)2. What is the difference between undefined and indeterminate form?3. Calculate the limit, citing each step you use along the way:lim x 3 (3x+ x2(2x+1)) 4. Calculate the following limits:a. lim x 2 (4-2x)/(x+2)b. lim x 0 (x2-1)/(x-1)c. lim x 0 (x 2 +1) x(x-1)d. lim x 0 x__ x2-x5. For the piecewise function, calculate lim x c- f(x), lim x c+ f(x), lim x c f(x)f(x)= 3x+2, if x<-1 c=-1 5+4x3, if x> -12.6 Continuity1. What is the definition of continuity at a point?2. T or F: f(x) must be both left hand and right hand continuous at x=c, if the function is continuous at x=c3. What are the three types of discontinuities, and their definitions?a.b.c.4. What is the delta-epsilon interpretation of continuity5. A function f(x) is called a _____________________________ if it is continuous on its domain6. Determine which type of discontinuity this has:lim x 2- f(x)=2, lim x 2+ f(x)=1, f(2)=17. Sketch the graph of the function with the listed propertieslim x 0- f(x)=-1, lim x 0+ f(x)=1, f(0)=08. Determine lim x c- f(x), lim x c+ f(x), lim x c f(x), and f(c). Is f continuous at c? If not, what type of discontinuity?F(x)= x 2 -2x-3, c=3 x-32.7 Two Theorems About Continuous Functions1. What is the Extreme Value Theorem?2. What is the Intermediate Value Theorem?3. A function can change signs only where on a graph?...i.e., if a function changes sign at x=c (from + to -, or vise versa), then f(x) is either _________ or _________ at x=c4. Use the EVT to show that f has both a maximum/minimum value on [a,b]. Then use a calculator to approximate these valuesf(x)=3-2x2+x3 , [0,2]5. For each f, a, and b given, the special case of the Intermediate Value Theorem may or may not apply. Determine if the theorem applies, and if so, use it to show that f has a root between x=a and x=b.f(x)=x3+x2-4x, a=1, b=26. Find the intervals on which f is positive and negative. Express answer in interval notation.F(x)= (x+4)(x-1) 2x+33.1 Tangent Lines and Derivatives at a Point1. What are the two types of definitions for a derivative?2. Find the derivative of the function y=x23. Using the other definition, find the exact slope of the tangent line y=x2 at x=-24. Find the equation of this tangent line using question 35. Find the slope of the curve y=1/x2 + 4x at x=1/26. Find the derivative of y=x3, at x=-17. Find the derivative of y=SQRT (x), at x=23.2 The Derivative of an Instantaneous Rate of Change1. The definition of the average rate of change is:f(b)-f(a) b-aChange this formula slightly to get the equation for the average velocity2. What is the definition for instantaneous velocity?3. For the function f on the interval [a,b], calculate the average rate of change of f from x=a to x=b.a. f(x)=1/x; [0.9,1.1]b. f(x)=(1-x)/(1+x3) [0,0.5]4. Find the instantaneous rate of change at f=c;a. f(x)=3x+1, c=-2b. f(x)=1/(x+1), c=-23.3 Differentiability1. What is the definition of differentiability at a point?2. What is the relationship between one-sided and two-sided differentiability?3. T or F: Does Differentiability imply continuity?4. T or F: Does Continuity imply differentiability?5. Determine if differentiable at x=c… if not, are they left-differentiable? Right differentiable?F(x)= 2x-5, c=-2F(x)= 1/x, c=06. Determine if (a) f is continuous at x=c, and (b) f is differentiable at x=cf(x)= -x-1, if x< -2 1-x2, if x> -2 c=-23.4 The Derivative as a Function1. Write the two definitions of the derivative of a function. How are these similar and yet different to the definition of a derivative at a point?2. Calculate the following derivatives using BOTH definitions of the derivative, then use your answer to calculate f’(-2), f’(0), and f’(3)a. f(x)= 3x+1b. f(x)=5c. f(x)= x3 + 2d. f(x)= x3+x3. Use the definition of the derivative to calculate the following derivatived 3 (2x3) dx34. Graph the function of y=x2 + x -6. Then graph the derivative of this function WITHOUT finding the derivative