1. Let f(x) = 5x². Show that f(3 + h) = 5h? + 30h + 45. Then show that f (3 + h) – f(3) = 5h + 30 and compute f'(3) by taking the limit as h → 0.

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

A: We know that, the difference quotient given as m=fa+h-fah            ⋯⋯1 is the slope of the secant…

Q: ´28 – e14z x 0 Determine whether the function f(x) = 14x – 1

A: Solution: Consider the given function is  f(x) = x8-e14x    x<014x-1       x≥0 To check whether…

Q: 3. If f is continuous and f (x) dx = 8, show that the graph of ƒ intersects the line y = 4 at least…

A: The function f is continuous. The integral function is ∫13fxdx=8.

Q: Is the limit of f (x) as x approaches c always equal to f (c)? Why or why not?

A: To prove :                   The limit of f(x) as x approaches c always equal to f(c). Answer :…

Q: For the function given by q (x)= x² + 2x+1, use the limit definition of the derivative to compute…

A: Definition of the Derivative:        The derivative of f(x)  with respect to x is the function f'(x)…

Q: x2 + 0x – 4 lim x-2 x - 2 Enter I for oo, -l for -co, and DNE if the limit does not exist.

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Q: Given f(x) = 4Va +5, find f' (x) using the limit definition of the derivative. %3D f'(1)=

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Q: Show that the function f(x, y) x4 + y? has no limit as (x, y) approaches (0,0).

A: Note: Since we can solve at most one problem at a time, we have solved the first one. Kindly…

Q: evaluate the limit:  limit of the square root of 1+t - square root of 1-t / t as t approaches 0

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Q: Express the limit as a definite in 7 lim 1 + (i/n)3 n→ 0 n i = 1 1 7n dx 8

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Q: f(x) = x(3x-(9x2+2)1/2). Compute the limit of f(x) as x tends to positive infinity.

A: Given data:   The expression for the given function is f(x)=x(3x-(9x2+2)1/2).   The expression for…

Q: Let            fn(x) = nx/1 + nx2  (a) Find the pointwise limit of (fn) for all x ∈ (0,∞).

A: Given : fn(x)=nx1+nx2; To find the point wise limit of this function for given values of x: We have…

Q: Compute the limit 1 a2 1 - x2 lim xa x а - What can you say about the derivative of f(x) = ?

A: Given limit :

Q: Use the limit definition of the derivative function to find 4 x2]. dz Which of the following sets up…

A: The objective is to use the definition of limit to find the derivative.

Q: Evaluate f x2 dx using the limit defintion of the definite integral.

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Q: Find the limit. 3 - ti + In(t)j k t-3

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Q: lim, - f{x)= 0, then f(x)dx diverges Select one: O True False

A: When the limit exist or equal to 0 then the series is convergent otherwise divergent.

Q: Suppose that f and g are discontinuous at x = c. Does it follow that f + g is discontinuous at x =…

A: Let us define that Then both f and g are discontinuous at x=2

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: Let f(x) = V9 – 2x. Find f'(x) using the limit definition of the derivative.

A: To find f'(x) by definition f(x)=√(9-2x)

Q: Which of following limits is the derivative of g(x) =5x at =2? 10 – 5h lim h 10 + 5h lim h>0 h 10 +…

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Q: S 1/x ifsxs1 n*x if 0sx< Given fn : [0, 1] → R, defined as f,(x) = then the point wise limit of…

A: Sequential limit

Q: Let f(x) = 3x² – 5x + 1 f(x+h)-f(x) (a) Find f'(x) using the limit definition of derivative. That…

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Q: The limit of f(x) = sin(x) as x approaches t is O T/x n/2

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Q: Show that the limit of the function f(x,y) = at (0,0) does not exist. (x+y)² x² + y²

A: Limit of a Function: Like a single-variable function, a multi-variable function can also have limit.…

Q: Find f'(a) using the limit definition for 1 f(x) = V6 – 5x -

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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: If f(2) is a continuous function at a point z, , then f(z) is differentiable at z, . But the…

A: Authoring guidelines:…

Q: Given x2 if x 2 f(x) – f(2) a) Determine the left and right limit of the function b) For what…

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Q: If [ f(x)\dx<∞ show that limf(x)dx=0 k→∞00 1

A: We have to prove that given expression.

Q: Find the limit of f(x) as x approaches 2, where

A: fx=1,  x≠20,  x=2

Q: Let fn f, where if 0<<! Sn(x) ={ S 2-a if 0<x<1 and f(x) if a = 0 Then lim sup fn (x) – ƒ (x)| =

A:

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: Show that f (x, y) = has no limit as (x, y)→(0,0) Vx2+y2

A: The objective is to show that f(x,y) has no limit as (x,y) approaches to (0,0).

Q: For every continuous f:[0,+0) →R _function , if a>0 2fx'f(x*)dx =[ xf(x)dx Demonstrate that this…

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Q: Let F(x) = | 3e“ dt. Find lim F(x) if it exists. If the limit does not exist, explain why. x→0

A: see below the answer

Q: 5. Consider the function S 2r – 1, si x < 2 f(x) = 6-x, si x 2 2. The Limit x 2 f(x) is:

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Q: Find the limit as x approaches negative infinity of (xe1/x-x)

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Q: 2. Find the (exact) values of a and b so that f is continuous at a = 2. 8r2 + 20x + 28 if x 2

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Q: Find the limit as x approaches -1 for O-1/2 O 1/2 O-1 01 x+2 -1 x+1

A: Concept: L'Hospital's Rule: A general technique for assessing indeterminate forms like 0/0 or ∞/∞ is…

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: 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: what is the limit of f(t) as t approaches 3. F(t)= ((1/t)-1/3)/t-3?

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Q: Suppose that f (x) is a continuous function. 3 If | f(x) dx=5, then what is the value of z - f(x)]…

A: Given, ∫13f(x) dx =5Now,∫13[x -f(x)] dx= ∫13x dx -∫13f(x) dx =x2213 -5…

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: Find the limit of (x3+x2)1/2sin(((2)1/2pi)/x) as x approaches 0

A: To find

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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Q: Compute the limit of the function at −1 AND determine if the function is continous at 1. f (x) = {…

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