(a) Using the formal definition of a limit, prove that f(x) = x³ - 2x is continuous at the point x = 1; that is, limx→1 (x³ - 2x) = = -1. (b) Draw a mapping diagram to illustrate your answer to (a) for the special case of € = = 1/2. (c) (i) Prove that: 77 ㅠ | sin (3x + 3x -|- 0.005 (ii) Using (i) prove that if x is within of, then sin(3x + 2) is approximately 0 correct to 2 decimal places. 3

Q: (a) By using the definition of limit, prove that lim (3x+i z31

A: Introduction: Limits in mathematics are distinct real numbers. Consider a real-valued function "f"…

Q: Find the limit lim x→∞ (1+5/x)x lim x→0 ((e2x-1-x)/x2)

A:

Q: 3. Describe the interval(s) on which the function is continuous. c) S(x)= x-1 2, x= 1

A:

Q: If f(x+ y)= f(x)f (y) for all x, y eR and f (x)=1+g(x)G(x), where lim g (x)=0 and lim G (x)…

A: The problem has been solved in detail in the next step.

Q: (-ト-c)s x-5

A:

Q: What is the length of EF? y 4 8 units 10- 7 units 4 2 units -10 -8 -6 4 10x -2- 6 units -4- -6 -8

A: solution of the given problem is below ...

Q: Use the Squeeze Theorem to fĩnd lim In(x + 1) cos(-)

A:

Q: Evaluate the limit algebraically: sin(2x) + cos(x) cos(x) lim

A: Given : limx→π3sin2x + cosxcosx

Q: Determine whether the limit lim xsin exist, 72 I

A:

Q: Let f(x) = Prove that lim f(x) = 0. x →0 sin x + sin 3x + sin 5x, if x is rational, 0, if x is…

A: We need to prove that limit is zero.

Q: Find lim f(x) and determine if continuous at x =2 エ→2

A: We have to check

Q: Consider the function: f(x) = x^3, if x ≤ 1, 12 − 5x, if 1 < x < 3, x^2 − 6x + 6, if x ≥ 3.…

A:

Q: Evaluate f(a), lim f(x), lim f(x), lim f(x). Is f(x) continuous at x = a. x→a- x→a+ xa if x 4

A: For 4^- we use the first piece of the function

Q: -{ e-2, Cos x, x 0

A:

Q: 3. g(x) x² +5

A:

Q: a) f(x) = x + √2x − 1

A:

Q: A function f is differentiable at a number a.. A. ... if lim f(x) = f(a), f(z) – f(a) B. ... if lim…

A: topic - condition for differentiable function