Wednesday, September 28, 2016

Unizor - Derivatives - Function Limits - Exercise 3

Notes to a video lecture on

Function Limit - Exercise

Try to do these exercises yourself.

All function limits below are supposed to be calculated as argument x tends to real number a.
In other words, we say that
f(x)→L as x→a, if
 positive ε (however small)
δ: (|x−a| ≤ δ) ⇒ (|f(x)−L| ≤ ε)

1. If x=a is a point of continuity for function f(x) then, as follows from the definition of continuous function,
f(x) → f(a)
Find the following limits:
3x³-2x²+x-1 as x→1
2x as x→3
sin(x) as x→π/2
lg(10x) as x→100

2. Indeterminate 0/0 or 0·∞
(x²−4)/(x−2) as x→2
(1−cos(x))/x² as x→0
(5x−1)]/x as x→0
sin(x)/(ex−1) as x→0

3. Indeterminate ∞−∞
sin(x)·tan(x) − sec(x) as x→π/2
1/(x²-3x+2) − 1/(x²+5x−6)
as x→1
ln(sin(x)) − ln(x) as x→0
x+1/(x−4) − √x+16/(2x−8)
as x→4

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