Could someone please solve these four problems with explanations? I'd like to understand how to get to the answers. Thank you!
Without using a calculator:
For each of the following, find:
I. lim x->a- f(x)
II. lim x->a+ f(x)
III. lim x->a f(x)
A. f(x)=|x^2+3x+2|/x^2-4 for a=2
B. f(x)= {sinx, if x<pi/6
{tanx, if x=pi/6
{cosx, if x>pi/6
for a=pi/6
C. f(x)= {sin x/3, if x≤pi
{x sqrt3/2pi, if x>pi
for a=pi
D. f(x) = (x^2-36)/sqrt(x^2-12x+36)
for a=6
A. the numerator is always positive, so since there is a vertical asymptote at x=2,
(i) -infinity
(ii) +infinity
(iii) not exist
B. f(x) is very conveniently defined, so that we have
(i) sin(pi/6) = 1/2
(ii) cos(pi/6) = √3/2
(iii) not exist
C. again a convenient definition for f(x)
(i) sin(pi/3) = √3/2
(ii) pi*√3/2 pi = √3/2
(iii) √3/2
D. x^2-12x+36 = (x-6)^2, so
f(x) = (x^2-36)/|x-6| =
{x+6 for x>6
{-(x+6) for x<6
(i) -12
(ii) 12
(iii) not exist
Sure! Let's go through each problem step by step, and I'll explain how to find the limits.
A. f(x) = |x^2 + 3x + 2| / (x^2 - 4) for a = 2
To find the limits, we need to evaluate the function for values of x approaching 2 from both the left and the right, as well as at x = 2 itself.
I. lim x->2- f(x):
This means we need to find the limit of f(x) as x approaches 2 from the left. Plug in values of x slightly less than 2 into the function and see what the expression evaluates to. For example, let's choose x = 1.9:
f(1.9) = |1.9^2 + 3(1.9) + 2| / (1.9^2 - 4)
Calculate this expression (|8.61 + 5.7 + 2| / (-2.79)), and you will get the value for the left-hand limit.
II. lim x->2+ f(x):
This means we need to find the limit of f(x) as x approaches 2 from the right. Plug in values of x slightly greater than 2 into the function and see what the expression evaluates to. For example, let's choose x = 2.1:
f(2.1) = |2.1^2 + 3(2.1) + 2| / (2.1^2 - 4)
Calculate this expression (|9.261 + 6.3 + 2| / (-3.79)) to find the value for the right-hand limit.
III. lim x->2 f(x):
This means we need to find the limit of f(x) as x approaches 2. Here we substitute x = 2 directly into the function and evaluate the expression:
f(2) = |2^2 + 3(2) + 2| / (2^2 - 4)
Calculate this expression (|4 + 6 + 2| / 0) to get the value for the limit.
B. f(x) = {...} for a = pi/6
Following the same steps as in problem A, we find the limits for this piecewise-defined function.
I. lim x->pi/6- f(x):
Evaluate f(x) for values of x slightly less than pi/6.
II. lim x->pi/6+ f(x):
Evaluate f(x) for values of x slightly greater than pi/6.
III. lim x->pi/6 f(x):
Evaluate f(x) for x = pi/6.
C. f(x) = {...} for a = pi
Again, follow the steps:
I. lim x->pi- f(x):
Evaluate f(x) for x slightly less than pi.
II. lim x->pi+ f(x):
Evaluate f(x) for x slightly greater than pi.
III. lim x->pi f(x):
Evaluate f(x) for x = pi.
D. f(x) = (...)/... for a = 6
Similarly, follow the steps:
I. lim x->6- f(x):
Evaluate f(x) for x slightly less than 6.
II. lim x->6+ f(x):
Evaluate f(x) for x slightly greater than 6.
III. lim x->6 f(x):
Evaluate f(x) for x = 6.
By using these steps, you'll be able to find the limits for each of the given problems. Good luck!