9. For b a positive constant, let A(b) be the area of the region enclosed by f(x) x² and g(x) = b. Let T(b) be the area of the triangle COD as shown in the figure. Find lim T(b) if it exists. If 6+0 A(b) the limit does not exist, explain why. לא=יa

Q: Evaluate the limit: lim_ (x - ln x)* x→0+

A: Using L Hospitals rule , Value of given limit is evaluated.

Q: Find the limit using L'Hospital's Rule. lim x→0+ \x еx — 1/

A:

Q: Suppose that f(x, 0) = 3 for all x ≠ 0 and f(0, y) = 5 for all y ≠ 0. What can you conclude about…

A: The limit of the function, f(x,y) at a point (a,b) does not exist if two paths upon which the…

Q: Find the limit lim Vz +1 Enter I for o0, -I for-oo, and DNE if the limit does not exist. Answer

A:

Q: Q4)A) Find the following limits (for two only). x+1 1. limx→-1 х2-х-2 1-Vx 2. limx→1 1-x 2х 3.…

A: Part 1,Given,           limx→-1x+1x2-x-2

Q: Show the limit does not exist. lim (1,y)→(0,0) x6 + y²

A:

Q: 6 — х — а? if x 1 - Calculate the following limits. Enter "DNE" if the limit does not exist. lim…

A:

Q: and If lim f(x) = lim g(x) = +o and flx) =1 and gtxl = e, what is the limit of the division of !!…

A: As given limit is infinitely by infinity form so use L'hopital's rule and substitute the given…

Q: Use definiction to १r० v of Conkegene to lim n+4

A:

Q: Evaluate the limit: lim (x - ln x)* x→0+

A:

Q: lim (x 1/2 ln x) X→∞

A:

Q: Evaluate the limit using L'Hospital's rule: e10x – 1– 10x - - lim x2

A:

Q: Find the limits as x 0 and as x→ -o. (If the limit is infinite, enter 'oo' or '-0', as appropriate.…

A:

Q: 4. lim ) ズ→1 Xー1

A:

Q: . lim ( 1)0o* 4x +

A:

Q: Find the limit L. Then using the Epsilon Delta definition to prove that the limit is L  lim x→−4…

A: Given,

Q: In finding the limit for  (x)2/x as x→+∞ by L'Hospital rule, which limit should appear in the late…

A:

Q: Evaluate the limit using L'Hospital's rule: x13* lim x→0 13* – 1 -

A:

Q: Use L'Hôpital's Rule to find the limit. 5. lim x0 e2- xr- 1 5 cos x

A: l'Hosptital's rule provides technique to evaluate the limit with indeterminate form.

Q: (c) Find the following limits WITHOUT using the L’Hopital's Rule. -5x2-8 lim x→+∞ 2x+3 (i) Зx+1-сos…

A: According to our guidelines we can answer only three subparts, or first question and rest can be…

Q: 8. Evaluate the limits by previous methods or by L'Hospital's Rule, if applicable: 4x3 +3x 4x b) lim…

A: We have to find limit

Q: . Suppose a state's income tax code states that tax liability is 12% on the first $20,000 of taxable…

A:

Q: If x4 <=ƒ(x)<= x2 for x in [-1, 1] and x2<=ƒ(x) <= x4 for x <-1 and x> 1, at what…

A: Summarizing the given information:             . We need to find out that at what points c, we…

Q: Find the limit, if it exists, or type N if it does not exist. 3x2 lim (z,y)→(0,0) 5x² + ly?

A: Solution: The objective is to find the limit

Q: Show that for a > 1/2, lims→0 sªW(1/s) = 0, where the limit is taken in the mean square sense.

A: Please find the step by step solution below. Thank you

Q: (b) Find the following limits WITHOUT using the L'Hopital’s Rule. x²–4x lim x→0- |x| (i) (ii) 2tan?x…

A: Part i,Given,           limx→0-x2-4xx

Q: Q4)A) Find the following limits (for two only). x+1 1. limy--17-x-2 1-VE 2. limy-1 1-x 2x 3. lim--0…

A:

Q: Evaluate the limit using L'Hospital's rule. limx→∞(x^5)(e^(−x^2))

A: Given limit is It is of indeterminate from (infinity/infinity),  Therefore, by applying L'Hospital…

Q: 2) Suppose that the limit of a function f(x) exists at z = a. If lim f(x) = M and lim f(x) = N, what…

A: Given:- f(x) exists at x=a We have to find the relation between M and N for given limit of a…

Q: gives What is the value of a, such that after direct substitution the limit lim (x)1+Inx Select one:…

A: Given: limx→∞xa1+lnx=∞0 Use limit rule: if limx→af(x)=b and limx→ag(x)=c then limx→af(x)g(x)=bc

Q: 2 — х — г? if x 5 - Calculate the following limits. Enter "DNE" if the limit does not exist. lim…

A: Here we will find the given limits,

Q: lim 0-4 In[(1 + h)² + 1] - ln(2) y

A: To evaluate the given limit .

Q: 2xy lim (x,y)→(0,0) x² + y²’

A:

Q: Q3 (A): By using "L' Hopital rales "; Find the limits of: (1) tim-al-ca (2) lim, x(tan"¹x-x)

A:

Q: Find the limits limx→0+x (ln x)2

A: Consider : limx→0+x(ln2x). Modify the limit function. limx→0+(ln2x)1x…

Q: 4. Find the following limits or determine that they do not exist: 1+1/x a. lim,-0 1+1/x² 1+1/x? b.…

A:

Q: 1. Use a table of values to estimate the value of the limit. ex – 1 lim x-0 tan x

A: Given: fx=ex-1tanx limx→0ex-1tanx   Note: For tanx, x should be in radian.

Q: . lim,0+ (1 + 2x)In(2z)

A: Solution :1∞  formif limx→0fx=1 limx→0gx=∞if the form limx→0 fxgx then elimx→0 fx-1gx

Q: show that L’Hôpital’s Rule is applicable to the limit as x →±∞and evaluate

A:

Q: 4. Use the € & definition of limit to show that lim (2z+3i) = 2 + 5i. z+1+i

A: As per our guideline, we are supposed to solve only first question. Kindly repost other questions as…

Q: Find the limit lim, →+∞ In(x² – 1)– In(x+ 1)| Enter I for x∞, -l for -o, and DNE if the limit does…

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Q: We determined that l'Hospital's rule applies to the limit  lim x→5   x2 − 10x + 25 x2 − 5x

A:

Q: Evaluate the limit sin 2x lim z+0 9x

A: Answer and explanation is given below

Q: x – 4 lim 244 x2 – 3x – 4

A:

Q: Use l’Hôpital’s rule to find the limits

A: Since, we have to find the limits of the function given below:-limh→0 eh-(1+h)h2 -(i)

Q: Use l’Hôpital’s rule to find the limits

A: Let the given limit be L - L = limx→0 x2lnsecxUsing L-hospital rule, differentiating numerator and…

Q: Compute the limit analytically using the technique of your choice: lim d) x→0 sin(bx)

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Q: Use l’Hôpital’s rule to find the limits

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Q: 5x2 + 10х + 1 :0. lim x4 + x? + 1

A:

Q: Find the limit. lim Cn (3+x)- In (l+x)]

A:

Minimization

In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.

Maxima and Minima

The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.

Derivatives

A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.  

Concavity

In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.