Question

mcv4u
problem set #1a: limits

1. given each table of values below:

i) state the limit which it can be used to determine.
ii) evaluate the limit, if it exists.
a)

$x$	$f(x)$
2.9997	21.9946
2.9998	21.9964
2.9999	21.9982
3	22
3.0001	22.0018
3.0002	22.0036
3.0003	22.0054
3.0004	22.0072

b)

$x$	$g(x)$
-5.0003	-10.0003
-5.0002	-10.0002
-5.0001	-10.0001
-5	dne
-4.9999	-9.9999
-4.9998	-9.9998
-4.9997	-9.9997
-4.9996	-9.9996

c)

$x$	$h(x)$
7.9997	-208.3372
7.9998	-312.5039
7.9999	-625.0039
8	dne
8.0001	624.9961
8.0002	312.4961
8.0003	208.3294
8.0004	156.2461

d)

$x$	$j(x)$
-6.01	-9.0603
-6.001	-9.0060
-6	18
-5.999	-8.9940
-5.99	-8.9403
-5.9	-8.4295

e)

$x$	$k(x)$
-2.0003	0.0075
-2.0002	0.005
-2.0001	0.0025
-2	0
-1.9999	0.0025
-1.9998	0.005
-1.9997	0.0075
-1.9996	0.01

f)

$x$	$m(x)$
100	-10.8988
1000	-10.9002
10000	-10.9110
100000	-10.9859
1000000	-10.9919
10000000	-10.9996

Explanation:

Part a)

Step1: Analyze left limit as $x\to3$

As $x$ approaches 3 from values less than 3, $f(x)$ approaches 22.

Step2: Analyze right limit as $x\to3$

As $x$ approaches 3 from values greater than 3, $f(x)$ approaches 22.

Step3: State the limit

Left and right limits match, so the limit exists.

Part b)

Step1: Analyze left limit as $x\to-5$

As $x$ approaches -5 from values less than -5, $g(x)$ approaches -10.

Step2: Analyze right limit as $x\to-5$

As $x$ approaches -5 from values greater than -5, $g(x)$ approaches -10.

Step3: State the limit

Left and right limits match, so the limit exists.

Part c)

Step1: Analyze left limit as $x\to8$

As $x$ approaches 8 from values less than 8, $h(x)$ approaches $-\infty$ (values grow negatively without bound).

Step2: Analyze right limit as $x\to8$

As $x$ approaches 8 from values greater than 8, $h(x)$ approaches $+\infty$ (values grow positively without bound).

Step3: State the limit

Left and right limits do not match, so the limit does not exist.

Part d)

Step1: Analyze left limit as $x\to-6$

As $x$ approaches -6 from values less than -6, $j(x)$ approaches 18.

Step2: Analyze right limit as $x\to-6$

As $x$ approaches -6 from values greater than -6, $j(x)$ approaches 18.

Step3: State the limit

Left and right limits match, so the limit exists.

Part e)

Step1: Analyze left limit as $x\to-2$

As $x$ approaches -2 from values less than -2, $k(x)$ approaches 0.

Step2: Analyze right limit as $x\to-2$

As $x$ approaches -2 from values greater than -2, $k(x)$ approaches 0.

Step3: State the limit

Left and right limits match, so the limit exists.

Part f)

Step1: Analyze limit as $x\to+\infty$

As $x$ increases without bound, $m(x)$ approaches -11.

Step2: State the limit

Values approach a consistent finite number, so the limit exists.

Answer:

a) $\lim_{x \to 3} f(x) = 22$
b) $\lim_{x \to -5} g(x) = -10$
c) $\lim_{x \to 8} h(x)$ does not exist (DNE)
d) $\lim_{x \to -6} j(x) = 18$
e) $\lim_{x \to -2} k(x) = 0$
f) $\lim_{x \to +\infty} m(x) = -11$

Limit Type Justification (for part i):

a) Two-sided limit (approaching 3 from both sides)
b) Two-sided limit (approaching -5 from both sides)
c) Two-sided limit (approaching 8 from both sides, DNE)
d) Two-sided limit (approaching -6 from both sides)
e) Two-sided limit (approaching -2 from both sides)
f) Limit at positive infinity (x grows without bound)