Question

f(x)=x^2-4/x-2,then show that limit exist​

Answer 1

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FORMULA TO BE IMPLEMENTED

CONDITION FOR EXISTENCE OF LIMIT

A function f(x) is said to have a limit L at a point x = a if

$\displaystyle \lim_{x \to a + } \: f(x) =\displaystyle \lim_{x \to a - } f(x) = L$

In this case we write

$\displaystyle \lim_{x \to a} f(x) = L$

TO FIND

$\displaystyle \lim_{x \to 2} \frac{x^2-4}{x-2}$

CALCULATION

RIGHT HAND LIMIT

$\displaystyle \lim_{x \to 2 + } \frac{x^2-4}{x-2}$

$= \displaystyle \lim_{x \to 2 + } \frac{(x + 2)(x - 2)}{x-2}$

$= \displaystyle \lim_{x \to 2 + } {(x + 2)}$

$= 2 + 2$

$= 4$

LEFT HAND LIMIT

$\displaystyle \lim_{x \to 2 - } \frac{x^2-4}{x-2}$

$= \displaystyle \lim_{x \to 2 + } \frac{(x + 2)(x - 2)}{x-2}$

$= \displaystyle \lim_{x \to 2 - } {(x + 2)}$

$= 2 + 2$

$= 4$

So

$\displaystyle \lim_{x \to 2 + } \frac{x^2-4}{x-2} = \displaystyle \lim_{x \to 2 - } \frac{x^2-4}{x-2} = 4$

RESULT

Hence

$\displaystyle \lim_{x \to 2} \frac{x^2-4}{x-2} = 4$

Answer 2

Answer:

FORMULA TO BE IMPLEMENTED

CONDITION FOR EXISTENCE OF LIMIT

A function f(x) is said to have a limit L ata point x = a if

\displaystyle \lim_{x \to a + } \: f(x) =\displaystyle \lim_{x \to a - } f(x) = Lx→a+limf(x)=x→a−limf(x)=L

In this case we write

\displaystyle \lim_{x \to a} f(x) = Lx→alimf(x)=L

TO FIND

\displaystyle \lim_{x \to 2} \frac{x^2-4}{x-2}x→2limx−2x2−4

CALCULATION

RIGHT HAND LIMIT

\displaystyle \lim_{x \to 2 + } \frac{x^2-4}{x-2}x→2+limx−2x2−4

= \displaystyle \lim_{x \to 2 + } \frac{(x + 2)(x - 2)}{x-2}=x→2+limx−2(x+2)(x−2)

= \displaystyle \lim_{x \to 2 + } {(x + 2)}=x→2+lim(x+2)

= 2 + 2=2+2

= 4=4

LEFT HAND LIMIT

\displaystyle \lim_{x \to 2 - } \frac{x^2-4}{x-2}x→2−limx−2x2−4

= \displaystyle \lim_{x \to 2 + } \frac{(x + 2)(x - 2)}{x-2}=x→2+limx−2(x+2)(x−2)

= \displaystyle \lim_{x \to 2 - } {(x + 2)}=x→2−lim(x+2)

= 2 + 2=2+2

= 4=4

So

\displaystyle \lim_{x \to 2 + } \frac{x^2-4}{x-2} = \displaystyle \lim_{x \to 2 - } \frac{x^2-4}{x-2} = 4x→2+limx−2x2−4=x→2−limx−2x2−4=4

RESULT

Hence

\displaystyle \lim_{x \to 2} \frac{x^2-4}{x-2} = 4x→2limx−2x2−4=4