limit x->1- tan(pi*x/2) — step-by-step solution

1) Compute the limit of

\tan (π \cdot \frac{x}{2})

x \to 1

⁻

The limit of

\frac{x}{2}

is (

\frac{1}{2}

)⁻ when

x \to 1

⁻

ⓘ The limit of a product of finite limits is the product of the limits

The limit of

π \cdot \frac{x}{2}

π \cdot \frac{1}{2}

when

x \to 1

⁻

⇥1.1) Let's calculate:

π \cdot \frac{1}{2}

⇥ⓘ Multiplying a fraction by a term multiplies its numerator:

π \cdot \frac{1}{2}

\frac{π \cdot 1}{2}

\frac{π}{2}

⇥

\frac{π}{2}

ⓘ To determine the side of approach to the pole, we study the difference between the argument and its limit■The limit of a sum of finite limits is the sum of the limits

The limit of

x - 1

is (

1 - 1

)⁻ when

x \to 1

⁻

⇥1.2) Let's calculate:

1 - 1

⇥ⓘ Start from

1

and move back

1

on the number line, down to zero:

1 - 1

0

⇥

0

ⓘ The limit of a product of finite limits is the product of the limits

The limit of

π \cdot (x - 1)

is (

π \cdot 0

)⁻ when

x \to 1

⁻

⇥1.3) Let's evaluate:

π \cdot 0

⇥ⓘ A product with a zero factor is zero:

π \cdot 0

0

⇥

0

The limit of

\frac{π \cdot (x - 1)}{2}

0

⁻ when

x \to 1

⁻

ⓘ This is a reference limit of the function tan

So the limit of

\tan (π \cdot \frac{x}{2})

+∞

when

x \to 1

⁻

▶Limit:

+∞