lim x cos - 1 = lim cos(1/x) x 00 1/x [- sin(1/x)](1/x²) - 1/x = lim = lim sin = 0

Q: L’Hôpital’s Rule is used incorrectly. Describe the error.

A: Given: limx→∞xcos1x

Q: lim (1-cos x1/2) / ln(1+x) ? x-->0+

A: Here we use the series expansion of the functions cos (x) and ln(1+x):

Q: 3x+7 sin2x x+x.cosx 1- lim- x2-2 lim 2x2+x 3 - lim- x→0 2 - 00+X x→0 sinx.cosx 1 lim x-1- 1-x2 x-3…

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A: Given, limx→∞1-cos1x To Find the value of this limit function.

Q: lim sin x x aproches infinity

A: We need to determine the value of the following limit

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Q: Use series to evaluate the limits in Exercises

A: Note: Since you have asked multiple question, we will solve the first question for you. If you want…

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A:

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A: Definition and methods help to find the required limit of the given function. A value is said to be…

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A: This question is based on limits of 0infinty indeterminate form.

Q: 1 lim x? sin 1-cos³ x lim X→0 sin² x csc? x-2 lim п cotx-1 4

A: L'hospital rule: If limx→cf(x)g(x)=00 or ∞∞, then limx→cf(x)g(x)=limx→cf'(x)g'(x). Limit formula:…

Q: tan lim x+1- cos-l x

A:

Q: Find the limit

A: We know that cos0=1  For x=0-, cosx will be less than 1.  So, |cosx-1|= -(cosx-1)= 1-cosx for x=0-…

Q: lim xS0 cos2x - cos x / x^2

A: To find: limx→0 cos2x-cos xx2

Q: limx→1 3 (x-1) tan(x³²-x) (x+1) sin²(2x − 2) -

A:

Q: lim x to (0) cosecx - cotx

A: Find the limit as follows:

Q: lim x----->0    sin x sin 2x/1- cos x

A: Given:-  limx→0  sinx  sin2x1-cosxTo find:-     value  of the above limit.Concept:-  Use 'L Hospital…

Q: X-T 2. Find lim X→T sin x х-sinx 3. Find lim x3 X→0 e3x- e 5x 4. Find lim tan x-x 5. Find lim x→0…

A: Consider the following limit: limx→πx-πsinx Use L'Hopital's Rule:…

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Q: sin (x – 4) x < 4 Let F (x) cos (x – 4) 4< x Find lim F (x). x4+ lim F (x) = x→4+ %3D

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Q: x²+3 1) lim X00 x2+4 1-cos x sin x 2) lim- X-0 sinx 3) lim X-0 ex-1

A:

Q: sin x-coS X Determine el límite: lim 1-cotx

A: The given expression is limx→π4sinx-cosx1-cotx. Evaluate the limit as shown below.

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Q: Lim x tends to pi/4- (LHL) √(1-√sin2x)/(pi - 4x)

A:

Q: evaluate the limit.

A: Given:

Q: ✓cos x lim x 0 1 cos x - -

A: The given limit is limx→01-cosx1-cosx

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A:

Q: 4.x (a) lim 10 sin 5x sin() (b) lim 5x sin 3x (c) lim 10 x2 tan 2.x

A: Read the following instructions once 1. Short method is provided at the end of the subpart (c)…

Q: lim x -> pi/4 sinx-cosx/1-tanx

A:

Q: lim x→0 (sin3x sin5x/x2) *without use of deritivatives or l'hospital's rule*

A: See the details solution in below

Q: Lim e^x +e^-x-x²-2/sin²x-x² when x approaches 0

A: The given limit is, This limit is in 0/0 form, so we can apply L’Hospital's rule,

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A: Given,           limx→02sinx-sin2xx-sinx

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A: Introduction: So, according to L'Hospital's Rule, if we have an indeterminate form 00 or ∞∞, all we…

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A: To find lim x -&gt; 0    ln(1+x)-x / 1-cosx   = ?

Q: sin (1 cos x) sin x -0" sin Vx sin (x2 + x) 25. lim 26. lim sin (sin x) 27. lim 28. lim sin (x - 4)…

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Q: x+xcosx A: lim x-0 sinx cosx tan2(3x) B: lim x-0 x² 1-tan?x C: lim zcosx-1)

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Q: lim x (1 – cos X→-0

A: limx→-∞x1-cos1x

Q: lim sin2x/ x            x---&gt;0 lim tanx/tan2x       x----&gt;0

A: Multiplied the numerator and denominator by 2When x-&gt;0 then 2x-&gt;0Answer: 2

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A: Consider the limit limx→02sin x-sin 2xx-sin x. On applying the limit, it gives the indeterminate…

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Q: Lim Sin 2x Sin 2x |- CoS1ox

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A: