Question

Let f(x) = 5x^{2} + x − 2. Use the **limit
definition** of f′(a) and the Limit Theorems only to show
that f′(2) exists and equals 21.

Answer #1

Let f(x, y) = x^3 − 4xy^2 , x, y ∈ R. Use the definition of
differentiability to show that f(x, y) is differentiable at (2,
1).

Use the limit definition of the derivative of function x=a,
f'(a) to find an equation of the tangent line to the curve
y-f(x)-sqrt(x+2) at x-1. That is,first you find f'(1) and then find
the equation of the tangent line.

3. Use the definition of a limit (Definition 3.1.1) to show the
following: (a) limx→2 (x 2 + 2x − 5) = 3 (b) limx→1 (x 3 + 2x + 1)
= 4 (c) limx→0 (x 3 sin(e x 2 )) = 0

State (precisely) the limit definition for the derivative of an
arbitrary function f(x) at the point
x0. Use the limit definition
to compute the derivative of the following
function: f(x) = 1/x^2

Assumptions:
The formal definition of the limit of a function is as follows:
Let ƒ : D →R with x0 being an
accumulation point of D. Then ƒ has a limit L at
x0 if for each ∈ > 0 there is a δ > 0
that if 0 < |x – x0| < δ and
x ∈ D, then |ƒ(x) – L| <
∈.
Let L = 4P + Q. when P = 6 and Q = 24
Define...

Let A be a set and x a number. Show that x is a limit-point of
A
if and only if there exists a sequence x1 ,
x2 , . . . of distinct points
in A that converges to x.

Use the formal definition of limit (epsilon-delta definition) to
prove that lim x--> 3 (x^2 + 5x) = 24.

Use the definition of the derivative to show that f(x) = |x^2 –
9| is not differentiable at x = 3

Using the definition of the derivative, find f'(x). let f(x) =
x^2/x+1

Show that the derivative of f(x) = 6+4x^2 is f' (x)=8x by using
the definition of the derivative as the limit of a difference
quotient.

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