31. Use the identity x + y y sin x + sin y = 2 sin cos 2 to prove that sin(h/2) cos h sin(a + h) – sin a = h h/2 | sin x Then use the inequality < 1 for x # 0 to show that |sin(a + h) – sin a| < |h| for all a. Finally, prove rigorously that lim sin x = sin a.

Q: 4. Find two possible non-linear functions & and such that goh= √sinx-x²

A: To find the non linear function by the given composition.

Q: Q2/ a) If f) = , what are the domain of f(x) and g(x)? sin 6x lim 0 sin 9x b) Find

A:

Q: Letr> 2. Show that sin² r - sin² 2 r-2 < 2. Justify your answer. Hint: Apply the Mean Value Theorem…

A: Let r&gt;2.   To Show: sin2 r-sin2 2r-2&lt;2.

Q: -1 3. Show that sin-¹x = tan X √1-x² if |x| < 1.

A: Introduction: When there are trigonometric functions present in an expression or equation,…

Q: 2 Using the relation 2 (1 cosx) x, V x € (0, - · 4)

A: A relation is given by, 21-cosx &lt; x2, where x≠0. The objective is to use the given relation to…

Q: Q2. Prove that sec2(tan-x) = 1+ x?

A:

Q: Show that the extreme values of f(x) = a sin x + bcos x tVa2 + b2. are

A:

Q: lim (tan x) cos x x-2 X-

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

Q: Q1\ Graph and find the domain and range of: a. F(x)=X+v4 – x2 b. X+Y2 =0

A: Note - according to our guidelines we can answer only  first question and rest can be reposted.I am…

Q: Find the limits

A:

Q: 16. Evaluate ft tan5 x sec¹ xdx.

A: We know that Sec²x = 1+ tan²x ∫ xn dx=( xn+1)/(n+1)

Q: et sin x — х - lim x → 0 5 cos x – 5 ||

A:

Q: 1. If f(x) = sinx, show that f(x + y)-f(x) = cos x siny - 2 sin x sin² 2 2Y-T

A: 1) Given f(x)=sin(x)

Q: 9) Use lim (a) lim sinx sin6x x+0 sin4x = 1 Find the limits. (b) lim 1-cos²x X-0 X

A: Answer (a)limx→0​sin4xsin6x​Multiplying and dividing the numerator by 6x and denominator by…

Q: Use the figures to evaluate each function given that f(x)=sinx,g(x)=cosx, and h(x)= tan x. g (α /2 )

A: gx=cosx. Since the x-coordinate of any point on the unit circle is the cosine value, therefore…

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: ifferentate: f(x) = COSX 1+sin x Apply the quotient ru

A: Topic:- application of derivatives

Q: If x sinx < 1 2 - 2 cos x hold for all values of x close to zero. Find limx¬0 xsinx ? 2-2cosx

A: 1-x26&lt;xsinx2-2cosx&lt;1 holds for all values of x close to zero. Now, limx→01-x26=1-0=1 &amp;…

Q: Prove that lim x-&gt;0+ √ x [1 + sin2 (2 π/x)] =0

A: Given: limx→0+x1+sin2 2πx By using the property of limit, the limit of a product is the product of a…

Q: by using mean value theorem prove |sin(x) – sin(y)I < ]x – y|

A: To prove sinx-siny≤x-y Let f(t)=sint ∀t∈(x,y) Then we have the mean value theorem f'(c)=f(x)-f(y)x-y…

Q: Let f be given by if x COS x Show that f is continuous everywhere.

A: Given: f(x)=sinx , if x&lt;π4cosx, if x≥π4

Q: B. Prove the following: NO Differentiation 4. lim = : -10 x → 25 5. lim x 0 6 lim 25-x √x-5 sin8x…

A:

Q: Show that sin x < x for all x ∈ (0, 2π).

A:

Q: Prove that: 3 (0) 2³ ( 1 - 4 ² ³ ; x ²) = + sin²¹ x (b) 2₁ (¹-x²) == tan²¹ x 1 H

A: