Give a pair of functions f : R –→ R and g : R → R such that g is not tangent |f (x)-g(x)| || to f at 0 and yet lim,0 eel = 0. Illustrate with a sketch.

Q: Let f(x) = sin(sin(sin.x)) for for all x in R (f(x) x-0 g (x), and g(x) = (sinx), VxER, then lim is…

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Q: Use the limit definition to compute f '(a) and find an equation of the tangent line.

A: Answer: f'(-2)=-1

Q: if r = -1 Is f(r) = · continuous at r= - -1? (Show your work using limits.)

A: Since you have asked multiple question, we will solve the first question for you as per our guide…

Q: Let f : [1, ∞) → R be a continuous function such that lim|f(x)| = 0. For n ≥ 1, let gn : [1, ∞) →…

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Q: If f(x, y) approaches 1 as (x, y) approaches (0, 0) along the path y = mx, where m is any real…

A: To check whether the given statement is true or false.

Q: x? sin(1/r) if r + 0 if x = 0. if x + 0 f (x) = %3D

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Q: compute f ' (a) using the limit definition and find an equation of the tangent line to the graph of…

A: Given f(x)=x2-x The limit definition for derivative of a function is given as: Using the above…

Q: Let f(x) = V13 – x. Compute f'(4) using the limit definition f'(4) =| Find an equation of the…

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Q: Let f(x) = 2x2 - 3x - 5. Show that the secant line through (2, f(2)) and (2 + h, f (2 + h)) has…

A: Definition used -                   The slope of secant line -…

Q: Let f(x) = V1– cos 2x a) f has a • discontinuity at 0. b) lim f(x) = c) lim f(x) =

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Q: Suppose h' is continuous x>0. IF x = 0 and h'(x) = 3√tan (2x)-1 it exists. i at Find h (0), 'F

A: For the solution of the problem follow the next step.

Q: B) Show that fx) = sin for x * 0. for x= 0. X = is not continuous.

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Q: use the limit definition to compute f'(a) and find an equation of the tangent line. f (x) = x3 + x,…

A: For a function f(x) the limit definition for derivative at x=a is f'(a)=limh→0f(a+h)-f(a)h. The…

Q: Use the Squeeze Theorem to calculate the limit. lim x sin

A: Given limit

Q: Use the Squeeze Theorem to fĩnd lim In(x + 1) cos(-)

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Q: Suppose f is continuous on [0, 1]. What is lim n→ ∞ 1 S fo 0 f(x) dx?

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Q: Evaluate the limits in Exercise 7 and 8. cos t dt sin 8. lim x x→0+ 7. lim

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: evaluate the limit  limit as x approaches infinity of x+cos(x)/ x+1

A: To find,

Q: Let f(x) be a continuous differentiable function that f(x)- 4 satisfies lim =D5 %3D x3 x-3 Which…

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Q: Use the limit definition to compute f '(a) and find an equation of the tangent line. f (x) = x3 + x,…

A: Anwer: f'(0)=1

Q: Express the limit as a definite integral. lim xk Ax [-3, 2] 4-0 k 1

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Q: evaluate the limit of xsin(1/x) as x approaches zero using the squeeze theorem.

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Q: Use the limit definition to compute f 1 (a) and find an equation of the tangent line. f(x) = 2x 2 +…

A: Given:                fx=2x2+10x,     a=3 Find: f3+h and f3 f3+h=23+h2+103+h…

Q: Show that the function given by f(x,y)= x² + 3xy is differentiable at every poin using the…

A: Introduction: Distinctiveness (differentiability) implies continuity. If a function is…

Q: (-cx² + 2x, (sin x + cos x, x> n x<π 9) Prove that lim (x* sin+5) = 5 10) f(x) = What value must c…

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Q: Use the Squeeze Theorem to prove lim = 0. x00 x+ e- For x 0, since 0 < e* <1, we have

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Q: Use the limit definition to compute f 1 (a) and find an equation of the tangent line. f (x) = x +…

A: For a function f(x) the limit definition for derivative at x=a is f'(a)=limh→0f(a+h)-f(a)h. For a…

Q: Give a pair of functions f: R →R and g : R →R such that g is not tangent to f at 0 and yet lim,0…

A: To give a pair of function f : ℝ→ℝ and g : ℝ→ℝ such that g is not tangent to f at 0 and yet limx→0…

Q: Let f(x) = 3√(x + 1) − 4 = (x + 1)1/3 − 4. Use the limit definition of the derivative to show that f…

A: Consider the given function: fx=x+13-4=x+113-4 Vertical tangent at point indicates that at that…

Q: Use the limit definition to compute f 1 (a) and find an equation of the tangent line. f (t) = 2…

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Q: Let f(x) = /20 – x. Compute f'(4) using the limit definition f'(4) - - Find an equation of the…

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Q: Estimate the limit numerically if it exists. (If an answer does not exist, enter DNE.) Vx - 2 lim x…

A: Given query is to find the limit of the expression.

Q: )Use the Sandwich Theorem to show that lim cos x-1 =0. x2 Find f'(x) for f(x) = sec(vx2 + 1).

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Q: Find the value of a for which the limit sin (ax) – sin x – x lim x→0 is finite and evaluate the…

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Q: Prove that lim vx sin (2) = 0 using the Squeeze Theorem. x-0+

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Q: Use the Squeeze Theorem to prove that lim væ[1+ (sin(27/x))²] = 0.

A: "Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: 7- Use the limit definition to compute f' (x) and find equation of the tangent line at r = 3. f (z)…

A: topic - derivative of a function using limits

Q: Evaluate using the Squeeze Theorem.

A: Given: limx→0 x sin 1x2  To evaluate: The limit using Squeeze theorem.

Q: Consider the function f (x) = x(Va – 1), where a > 0, x > 0. If 0 < a < 1, show that limx0f (x) is a…

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Q: Зу Let flx,y) = , find the lim(xy)¬(-1,1)( Зу %3D x+y' ) along the curve y=x² .

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Q: Let f (x) = xex . Use the limit definition to compute f ,(0), and find the equation of the tangent…

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Q: Find the limit: lim f(x) as x approaches 0 of Kx2 - tan2 (mx)cos(mx)/x2     (m not =0)

A: Consider

Q: Does the existence and value of the limit of a function f(x) as x approaches a ever depend on what…

A: Yes, existence and value of the limit of a function f(x) as x approaches a depends on what happens…

Q: d. Use the limit definition of the derivative, f (x) = lim f (x+h)-f(x) 1 +h)=j, to prove that VT =…

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Q: If f' is continuous on [0, ∞) and limx→∞ f (x) = 0 , showthat

A: Given, If f' is continuous on the interval [0,∞) and limx→∞f(x)=0 show that, ∫0∞f'(x)dx=-f(0)

Q: if r =-1 A if z + -1 2. Is f(r) = continuous at r =-1? (Show your work using limits.)

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: