**How to Sketch a Graph of a Function With Limits :**

Here we are going to see how to sketch a graph of a function with limits.

**Question 1 :**

Sketch the graph of a function f that satisfies the given values :

f(0) is undefined

lim_{ x -> 0} f(x) = 4

f(2) = 6

lim _{x -> 2} f(x) = 3

**Solution :**

From the given question,

- We understood that the functions is undefined when x = 0.
- When the value of x approaches 0 from left hand side and right hand side, limit value will approaches to 4.
- When x = 2, the value of y will be 6.
- When the value of x approaches 2 from left hand side and right hand side, limit value will approaches to 3.

The value of x approaches from left and right, the limit will approach the value 4.

When x approaches 2 from left and right, the limit will approaches to 3. The picture given above will illustrate the condition.

When x = 2, f(x) that is the value of y will be 6.

Hence the picture given above is the required graph of the statements given.

**Question 2 :**

Sketch the graph of a function f that satisfies the given values :

f(-2) = 0

f(2) = 0

lim_{ x -> -2} f(x) = 0

lim_{ x -> 2} f(x) does not exists

**Solution :**

From the given question,

When x = -2, the value of y will be 0.

When x = 2, the value of y will be 0.

When x tends to 2, the function does not exist. To show this, we have to show the graph with different values of y.

**Question 3 :**

Write a brief description of the meaning of the notation lim _{x -> 8} f(x) = 25

**Solution :**

When x approaches from left side and right side, the value of limit will approaches 25.

lim |
lim |

**Question 4 :**

If f(2) = 4, can you conclude anything about the limit of f(x) as x approaches 2?

**Solution :**

The given statement represents, when x = 2, the value of y will be 4.

**Case 1 :**

When x approaches from left side and right side, we will get same values approximately, or

**Case 2 :**

When x approaches from left side and right side, we will get different values.

Hence, we cannot conclude anything about the limit of f(x) as x approaches 2?

**Question 5 :**

If the limit of f(x) as x approaches 2 is 4, can you conclude anything about f(2)? Explain reasoning.

**Solution :**

Given :

lim _{x -> 2} f(x) = 4

From this, we may understand that

lim |
lim |

when x approaches 2 from left side and right side, the limit will approaches to 4.

Hence we cannot conclude anything about f(2).

**Question 6 :**

Evaluate :

lim x->3 (x^{2} -9)/(x - 3) if it exists by finding f(3^{+}) and f(3^{-})

**Solution :**

= im _{x->3} (x^{2} -9)/(x - 3)

= im _{x->3} (x + 3)(x - 3)/(x - 3)

= im _{x->3} (x + 3)

im = 6 |
im = 6 |

**Question 7 :**

Verify the existence of lim _{x -> 1} f(x)

**Solution :**

If the limit x -> 1 exists, then

lim _{x-> 1}- f(x) = lim _{x-> 1}+ f(x)

f(x) = (x - 1)/(x - 1) f(x) = 1 lim |
f(x) = -(x - 1)/(x - 1) f(x) = -1 lim |

Since lim _{x-> 1}- f(x) ≠ lim _{x-> 1}+ f(x), the limit does not exists.

After having gone through the stuff given above, we hope that the students would have understood, "How to Sketch a Graph of a Function With Limits"

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