# EVALUATE THE LIMITS FROM THE GIVEN GRAPH

## Evaluating Left hand and right hand limit

Problem :

Evaluate the following :

1) lim x->-1-   f(x)

2) lim x->-1+   f(x)

Solution :

1) lim x->-1-   f(x) = 3

2) lim x->-1+   f(x) = 1

## When the limit does not exist ?

There are three situations

1) Jump discontinuity

2) there is a vertical asymptote

3) there is a violent oscillation

Problem :

Evaluate the following :

1) lim x->-2-   f(x)

2) lim x->-2+   f(x)

3) lim x->-2  f(x)

Solution :

1) lim x->-2-   f(x)

Approaching -2 from left side, we get the value of y as 1. So,

lim x->-2-  f(x) = 1

2) lim x->-2f(x) = 4

Approaching -2 from right side, we get the value of y as 4. So,

lim x->-2+  f(x) = 4

3) lim x->-2  f(x)

Both left hand and right hand limits are not equal, the limit does not exists at x = -2.

lim x->-2  f(x) = DNE

## Limit does not exists when vertical there is vertical asymptote :

lim x->2-  f(x) = -inifnity

lim x->2+  f(x) = inifnity

lim x->2 f(x) = does not exists

## When the limit become undefined ?

lim x->4-  f(x) = 0

lim x->4+  f(x) = inifnity

lim x->4 f(x) = it is not a defined value, so undefined.

Problem 1 :

Use the graph of the function f(x) to answer each question. Use ∞, −∞ or DNE where appropriate.

 (a) f(0) =(b) f(2) =(c) f(3) =(d) lim x→0− f(x) (e) limx→0 f(x) = (f) lim x→3+ f(x) = (g) limx→3 f(x) =(h) lim x→−∞ f(x)

Solution :

(a) f(0) :

The curve does not pass through any points on the y-axis. So, the answer is does not exists.

(b) f(2) :

The curve is passing through (2, 0). So, the value of f(2) is 0.

(c) f(3) :

We see the filled circle at (3, 3). So, the value of f(3) is 3.

(d) lim x→0f(x) :

While approaching the value 0 from left side, the output becomes −∞. So, lim x→0f(x) = −∞.

(e) limx→0 f(x) :

At exactly x approaches 0, the output is 2.

limx→0 f(x) = 2.

(f) lim x→3+ f(x) :

Approaching 3 from right side, the output is 2. So, lim x→3+ f(x) = 2

(g) limx→3 f(x) :

Approaching 3 from left side, we get 1. Approaching 3 from right side, we get 2. On both sides, we get different value. So, limx→3 f(x) does not exists.

(h) lim x→−∞ f(x)

At y = 1, we see the horizontal asymptote, x approaches infinity. So, the required output at x→−∞ is 1.

lim x→−∞ f(x) = 1

Problem 2 :

Use the graph of the function f(x) to answer each question. Use ∞, −∞ or DNE where appropriate.

 (a) f(0) =(b) f(2) =(c) f(3) = (d) limx→−1 f(x) =(e) limx→0 f(x) =(f) lim x→2+ f(x) =(g) limx→∞ f(x)

Solution :

By observing the graph above, at y = 1, we see horizontal asymptote. We see vertical asymptotes at x = -1 and x = 2,

(a) f(0) :

When the input is 0, the output also is 0. So, f(0) = 0

(b) f(2) :

At x = 2, we have vertical asymptote, we dont know where the curve is going to intersect. Then value of f(2) does not exists.

(c) f(3) :

The curve is passing through the point (3, 0). So, the value of f(3) is 0.

(d) limx→−1 f(x) :

Approaching x = -1 on either sides, from left side it approaches + infinity, from the right side it approaches - infinity. Since they are not equal, limx→−1 f(x) does not exists.

(e) limx→0 f(x) :

Approaching 0 from either sides, we see that it gives 0. So, the value of limx→0 f(x) = 0

(f) lim x→2+ f(x) :

Approaching 2 from right side, it approaches - infinity.

(g) limx→∞ f(x) :

When x approaches infinity, the output is 1. So, limx→∞ f(x) = 1

Problem 3 :

The graph of a function f is drawn above, answer the questions:

 a) f(−4) = b) lim x→−4− f(x) =c) lim x→−4+ f(x) =d) limx→−4 f(x) =e) f(−2) =f) lim x→−2− f(x) = g) lim x→−2+ f(x) =h) limx→−2 f(x) =i) f(0) =j) lim x→0− f(x) = k) lim x→0+ f(x) =l) limx→0 f(x) =m) f(2) =n) lim x→2− f(x) =o) lim x→2+ f(x) =p) limx→2 f(x) =q) f(4) =r) lim x→4− f(x) =s) lim x→4+ f(x) =t) limx→4 f(x) =

Solution :

 a) f(−4) = 2b) lim x→−4− f(x) = 1c) lim x→−4+ f(x) = 1d) limx→−4 f(x) = 1e) f(−2) = 4f) lim x→−2− f(x) = 5g) lim x→−2+ f(x) = 4h) limx→−2 f(x) = DNEi) f(0) = undefinedj) lim x→0− f(x) = 4 k) lim x→0+ f(x) = 2l) limx→0 f(x) = DNEm) f(2) = 0n) lim x→2− f(x) = 0o) lim x→2+ f(x) = -2p) limx→2 f(x) = DNEq) f(4) = undefinedr) lim x→4− f(x) = -1s) lim x→4+ f(x) = + infinityt) limx→4 f(x) =  DNE

## Recent Articles

1. ### Finding Range of Values Inequality Problems

May 21, 24 08:51 PM

Finding Range of Values Inequality Problems

2. ### Solving Two Step Inequality Word Problems

May 21, 24 08:51 AM

Solving Two Step Inequality Word Problems