Download Limits and Continuity in Calculus I and more Lecture notes Latin in PDF only on Docsity!
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
Limits & Continuity
In this chapter, we’ll define how limit of function values are defined and calculated.
Definition: the limit of f(x) as x tends to a is defined as the value of f(x) as x
approaches closer and closer to a without actually reaching it and denoted by:
lim f(x) L x a
→
L is a single finite real number
It’s important to know
- We don’t evaluate the limit by actually substituting x = a in f(x) in general,
although in some cases its possible.
- The value of the limit can depend on which side its approach
- The limit may not exist at all.
Example 13: to explain the concept of limit, take the function f(x) = 2x – 4 if the
lim f(x) 2 * 1 4 2 x 1
→
But the following table express many values of x can be expressed close to 1.
Question: Why we take values approaches to 2 in example 13 instead we take x = 1
directly?
Solution: the answer about this question can be expressed in the following example:
f 3 1
x^2
1
( x)= +
−
If x = 0 then 1/0 = ∞
So..
x 0.5 0.8 0.9 0.99 0.999 1.001 1.01 1.1 1.
f(x) - 3 - 2.4 - 2.2 - 2.02 - 2.002 - 1.998 - 1.98 - 1.8 - 1.
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
In limits we avoid ∞
THE TANGENT PROBLEM
The word tangent is derived from the Latin word tangens, which means “touching.”
Thus a tangent to a curve is a line that touches the curve. In other words, a tangent
line should have the same direction as the curve at the point of contact. How can this
idea be made precise?
For a circle we could simply follow Euclid and say that a tangent is a line that
intersects the circle once and only once as in Figure (a). For more complicated curves
this definition is inadequate as shown in Figure (b)
Example 1 4 : Find an eq. of the tangent line to the parabola y = x
2 at point (1,1)?
Solution
x ±0.2 ±0.5 etc..
f(x) 1.00000 1.
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
moment, but how is the “instantaneous” velocity defined? Let’s investigate the
example of a falling ball.
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
Example 15: Discuss the function
x 3
x 9 f
2
( x) −
If (1) x = 1, x = 2
(2) x = 3
(3) x 1, x 2
(4) x 3
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
Example 16: find the limits of the following functions:
x 3
x 9 lim
2
x (^3) −
→
= lim(x 3 ) 3 3 6 (x 3 )
(x 3 )(x 3 ) lim x 3 x 3
→ →
x 2 x 2 2
(x 2 ) 4 lim
x 2 2
x 2 2
x 2
x 2 2 lim x 2
x 2 2 lim x 2 x 2 x (^2) − + +
→ → →
5 x 7 x 1
3 x 2 x 1 lim 2
2
x (^) + +
→
÷ x
2
x
x
x
x
lim
2
2
x
→
Note:
n m
n m b
a
0 n m
b x a x ...... b
ax a x ...... a lim
o
m 1 m 1
m m
o
n 1 n 1
n n
x
− −
− −
→
Example 17: find
(x 1 ) x 2 x 3
x 1 lim 2
2
x 1 − + +
→
Solution:
lim(x 1 ) lim x 2 x 3 2 lim(x 2 x 3 )
(x 1 ) x 2 x 3
(x 1 )(x 1 ) lim
2
x 1
2
x 1 2 x 1 x 1
→ → → →
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
Theorem I If g(x) ≤ f(x) ≤ h(x) and
lim g limh(x) L x a
(x) x a
→ →
Theorem II
(x a)
sin(x a) 1 orlim θ
sin θ lim θ 0 x a
→ →
Example 18:
. 7 x 7 x
sin 7 x
. 5 x 5 x
sin 5 x
sin 7 x
sin 5 x lim x 0
→
(x 2 )
(x 2 )(x 2 )
sin(x 2 ) lim x 4
sin(x 2 ) lim x 2 2 x 2
→ →
Left and right – side limits
Example 19: Discuss the
8 x x 2
4 x 2
3 x 2 x 2
lim f(x)if f(x) x 2 −
→
Solution:
If x > 2
Then f (
) = lim f(x) lim( 8 x) 8 2 6 x 2 x 2
→ →
If x > 2 then
f(
- ) lim f(x) lim( 3 x 2 ) 8 x 2 x 2
→ →
Then right limit ≠ left limit at x = 2
Then, we say that
lim f(x)
x→^2 doesn’t exists
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
Example 20: Find
x
lim x →
and x
lim x →−
Solution:
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
Infinite limits and Vertical Asymptotes
As the line x = a is a vertical asymptote if at least one of the following statements is
true:
Example 21:
Continuity
If the limit of a function as approaches can often be found simply by calculating the
value of the function at. Functions with this property are called continuous at a.
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
Solution:
Example 23: Where are each of the following functions discontinuous?
Solution:
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
section. This special type of limit is called a derivative and we will see that it can be
interpreted as a rate of change in any of the sciences or engineering.
Example 25: Find an equation of the tangent line to the parabola y =x
2 at point P(1,1).
Solution:
Using the point-slope form of the equation of a line, we find that an equation of the
tangent line at (1,1) is
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
Note:
Example 26: Find an equation of the tangent line to the hyperbola y = 3/x at point
Solution:
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
RATES OF CHANGE
is called the average rate of change of y with respect to x over an interval
Example 2 8 : A manufacturer produces bolts of a fabric with a fixed width. The cost of
producing x yards of this fabric is C= f(x) dollars,
College of Engineering By Group of Calculus I
Department(s): Elec. Eng. Dep. Phase: 1
. Semester I (201 8 - 2019 )
(a) What is the meaning of the derivative f’(x), what are its units?
(b) In practical terms, what does it mean to say that f’(1000) = 9?
(c) Which do you think is greater f’(50) or f’(500), what about f’(5000)?
Solution:
