Question

lim x=2 (x^2-7x+10)/(x-2)​

Answer 1

Solution!!

$\sf \to \displaystyle \lim_{x\to 2} \left(\dfrac{x^{2}-7x+10}{x-2}\right)$

Let's evaluate using transformations.

Evaluate the limits of numerator and denominator separately.

$\sf \to \displaystyle \lim_{x\to 2} (x^{2}-7x+10)$

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

Evaluate the limit.

$\sf \to (2^{2}-7(2)+10)$

$\sf \to (2-2)$

$\sf \to (4 - 14 + 10)$

$\sf \to (0)$

$\sf \to (0)$

$\sf \to (0)$

Since, the expression 0/0 is an indeterminate form, try transforming the expression.

$\sf \to \displaystyle \lim_{x\to 2} \left(\dfrac{x^{2}-7x+10}{x-2}\right)$

Write -7x as a difference.

$\sf \to \displaystyle \lim_{x\to 2} \left(\dfrac{x^{2}-2x-5x+10}{x-2}\right)$

Factor out x from the expression.

$\sf \to \displaystyle \lim_{x\to 2} \left(\dfrac{x(x-2)-5x+10}{x-2}\right)$

Factor out -5 from the expression.

$\sf \to \displaystyle \lim_{x\to 2} \left(\dfrac{x(x-2)-5(x-2)}{x-2}\right)$

Factor out (x - 2) from the expression.

$\sf \to \displaystyle \lim_{x\to 2} \left(\dfrac{(x-2)(x-5)}{x-2}\right)$

Reduce the fraction with (x - 2).

$\sf \to \displaystyle \lim_{x\to 2} (x-5)$

Evaluate the limit.

$\sf \to 2-5$

$\boxed{\sf \to -3}$