The graph of f(x) = sin(2x)/x is shown in Figure 20. Is the function f(x) continuous at x = 0? Why or why not?

Expert Solution

Definition of Continuity:

A function f (x) is continuous at x = a provided all three of the following conditions hold true:

• Condition 1: f (a) exists.

• Condition 2: limx→af(x) exists at x = a .

• Condition 3: limx→af(x) = f(a)

If a function f (x) is not continuous at x = a, the function is discontinuous at x = a .

Consider the following graph drawn using Maple.

The function f(x) = sin(2x)/x is discontinuous at x = 0, since the function is not defined for x = 0. The condition 1 is violated. At x = 0, the graphing utility evaluates x→0f(x) = limx→0sin(2x)/x = 2.

Junaid answered 1 year ago

1) The graph of the derivative f ' of a continuous function f is shown. (Assume...

1) The graph of the derivative f ' of a continuous function f is shown. (Assume the function f is defined only for 0 < x < ∞.) (a) On what interval(s) is f increasing? (2a) On what interval(s) is f decreasing? (b) At what value(s) of x does f have a local maximum? (2b) At what value(s) of x does f have a local minimum? x=? (c) On what interval(s) is f concave upward? (2c) On what interval(s) is...

find the line tangent to graph of f(x) at x=0 1) f(x)=sin(x^2+x) 2)f(x)=e^2x+x+1

For the following exercises, determine whether or not the given function f is continuous everywhere....f(x) = 2x + 5/x

For the following exercises, determine whether or not the given function f is continuous everywhere. If it is continuous everywhere it is defined, state for what range it is continuous. If it is discontinuous, state where it is discontinuous.f(x) = 2x + 5/x

f(x)=0 if x≤0, f(x)=x^a if x>0 For what a is f continuous at x = 0...

f(x)=0 if x≤0, f(x)=x^a if x>0 For what a is f continuous at x = 0 For what a is f differentiable at x = 0 For what a is f twice differentiable at x = 0

Find the Fourier series for the function on the interval (-π,π) f(x) = 1 -sin(x)+3cos(2x) f(x...

Find the Fourier series for the function on the interval (-π,π) f(x) = 1 -sin(x)+3cos(2x) f(x )= cos2(x) f(x) = sin2(x)

Find f(x) for the following function. Then find f(6), f(0), and f(-7). f(x)=-2x^2+1x f(x)= f(6)= f(0)=...

Find f(x) for the following function. Then find f(6), f(0), and f(-7). f(x)=-2x^2+1x f(x)= f(6)= f(0)= f(-7)=

Where is the function ln(sin x ) defined and continuous?

Find the 5th Taylor polynomial of f(x) = 1+x+2x^5 +sin(x^2) based at b = 0.

Consider the function on the interval (0, 2π). f(x) = sin(x) cos(x) + 2 (a) Find...

Consider the function on the interval (0, 2π). f(x) = sin(x) cos(x) + 2 (a) Find the open interval(s) on which the function is increasing or decreasing. (Enter your answers using interval notation.) increasing ( ) decreasing ( ) (b) Apply the First Derivative Test to identify all relative extrema. relative maxima (x, y) = (smaller x-value) (x, y) = ( ) (larger x-value) relative minima (x, y) = (smaller x-value) (x, y) = ...

1. Consider the following function F(x) = {2x / 25 0<x<5 {0 otherwise a) Prove...

1. Consider the following function F(x) = {2x / 25 0<x<5 {0 otherwise a) Prove that f(x) is a valid probability function. b) Develop an inverse-transformation for this function. c) Assume a multiplicative congruential random number generator with parameters: a: 23, m: 100, and xo: 17. Generate two random variates from the function for (x).

Question