limit as x approaches 3 of (3-x)/(|x-3|) — step-by-step solution | Explain MAths

limit as x approaches 3 of (3-x)/(|x-3|)

⚠ |x-3| is read as the absolute value abs(x-3)

Compute the limit of

\frac{3 - x}{abs (x - 3)}

as

x \to 3

The limit of

3 - x

is

3 - 3

when

x \to 3

⇥Let's calculate:

3 - 3

⇥

0

The limit of

x - 3

is

3 - 3

when

x \to 3

⇥Let's evaluate:

3 - 3

⇥

0

The limit of

abs (x - 3)

is

abs (0)

when

x \to 3

⇥Let's calculate:

abs (0)

⇥

0

For

\frac{3 - x}{abs (x - 3)}

, the limit takes the indeterminate form 0/0 when

x

→

3

On the left,

x < 3

, so

abs (x - 3)

=

- (x - 3)

and the quotient is

1

On the right,

x > 3

, so

abs (x - 3)

=

x - 3

and the quotient is

-1

The left and right limits are different: the limit does not exist

So the limit of

\frac{3 - x}{abs (x - 3)}

does not exist when

x \to 3

▶Limit:

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