Unit 2 written assignment questions and answer PDF

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Unit 2 topic on Limit of Function practice questions...

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Written Assignment 2

Limit of Function University of the People MATH 1211: Calculus Tatiana Peisl (Instructor) 21st April, 2021

Written Assignment 2

lim ¿

1. Find the limit:

(1+x)1/x

x→ 0

In limit of the exponential function (1+x)1/x as x approaches 0 is equal to e. since the value of e in infinite exponential function as x approaches 0 is the same. Since it is 1∞ form then the formula 1 ( ) x 1

¿ [1+ (1+x)1/x = lim x→ 0

=

lim ¿

=

lim ¿

[1 +

x→ 0

1+

x→ 0

∴

lim ¿

1 1 1 1

+

(1−x ) + 2 1 2

+

(1+x)1/x

x→ 0

x+

+

1 3

1 1 1 ( −1)( −2) 3 x x x x + ……+] 3

1 1 ( −1) 2 x x x + 2

(1−x )(1−2 x) + ……+ ] 3 + …..+

=e

=e

lim sin 2θ 2.

=2

θ →0

θ 0 form the L’Hospital rule can be use 0

since it is

lim 2 cos(2θ) θ →0

1

lim sin 2θ =2

θ →0

θ

lim 3. Find the limit:

y→ ∞

√ y 2 +2

5 y−6

√

lim 1+ y→ ∞

5−

=

6 y

lim ¿ y →∞

2 y 2 First divide by the highest denominator

1 6 5− y

=2

Written Assignment 2

lim 1 =

y→ ∞

5 lim Hence

y→ ∞

√ y2 +2

1 5

=

5 y−6

lim x−2 4. Find the limit: :

x →0

ǀ xǀ−2 =

x−2 −x−2 lim ¿

x → 0−¿

=

−2 −2

=

x−2 x−2

=1

¿

x−2 (x )−2 lim ¿

x→0 and

+¿

=1

¿

lim x−2 since LHL = RHL = 1

then

=1

x →0

ǀ xǀ−2

5. Find the value of k that would make the lim exist:

2 lim x +kx−10 x→2

x−2 x-2 in the denominator should cancel. This happens when

lim(x −3)( x+5 ) x→2

x−2 simplify

lim x 2 +5 x−2 x −10

=

x→ 2

Therefore, the value of k is 3:

lim x 2 +3 x−10 x→ 2

lim x 2+3 x −10 x→2

x−2

6. Find the intervals on which the function is continuous: Factors x2 -12x + 32 is (x-4) (x-8) =

x+ 4 (x−4)( x−8 )

The function has infinite discontinuity at x=4 and x=8.

x +4 x 2−12 x +32

Written Assignment 2

∴

The function continuous on the intervals is (-∞, 4) ∪ (4,8) ∪ (8+ ∞)

7. Find the intervals on which the function is continuous, f(x) = Expand (x+3)2 +6 = x2 + 6x +15

4 (x +3 )2 +6

4 =

2

x +6 x +15

There is no real root in the denominator so the function do not have infinite discontinuity on the set of real numbers. ∴

The function is continuous on the interval (-∞ , ∞)

8. Find the points of discontinuity for the function, f(x)= of discontinuity.

√ 6 x+3 . Identify the type

The function is continuous only within the domain of f(x). Domain of f(x) 6x +3 ≥ 0 x ≥−

1 2

The function f(x) has an endpoint discontinuity.

Written Assignment 2

9. The graph below shows the number of tuberculosis deaths in the United States from 1989 to 1998. Estimate the average rate of change in tuberculosis deaths from 1991 to 1996.

Average rate of change =

f ( 1996) −f (1991) 1996 −1991

Average rate of change =

1200 −1700 5

Average rate of change = -100

=

−5 00 5

= -100

death year

References f%5Cleft(x%5Cright)%3D%5Csqrt%7B6x%20%2B%203%7D - Limit Calculator Symbolab. (n.d.). Www.symbolab.com. Retrieved April 21, 2021, from https://www.symbolab.com/solver/limit-calculator/f%5Cleft(x%5Cright)%3D%5Csqrt %7B6x%20%2B%203%7D

Herman, E. & Strang, G. (2020). Calculus volume 1. OpenStacks. Rice University...