Question

evaluate x^45-1/x^40-1 limx->1​

Answer 1

SOLUTION

TO EVALUATE

$\displaystyle \lim_{x \to 1} \: \: \frac{ {x}^{45} - 1 }{ {x}^{40} - 1 }$

FORMULA TO BE IMPLEMENTED

We are aware of the formula on limit that

$\displaystyle \lim_{x \to a} \: \: \frac{ {x}^{n} - {a}^{n} }{ x - a} = n \: {a}^{n - 1}$

EVALUATION

$\displaystyle \lim_{x \to 1} \: \: \frac{ {x}^{45} - 1 }{ {x}^{40} - 1 }$

$= \displaystyle \lim_{x \to 1} \: \: \frac{ \frac{ {x}^{45} - 1}{ x - 1} }{ \frac{{x}^{40} - 1}{x - 1} }$

$= \displaystyle \: \: \frac{\displaystyle \lim_{x \to 1} \frac{ {x}^{45} - 1}{ x - 1} }{\displaystyle \lim_{x \to 1} \frac{{x}^{40} - 1}{x - 1} }$

$= \displaystyle \: \: \frac{\displaystyle \lim_{x \to 1} \frac{ {x}^{45} - {1}^{45} }{ x - 1} }{\displaystyle \lim_{x \to 1} \frac{{x}^{40} - {1}^{40} }{x - 1} }$

$= \displaystyle \frac{45 \: \times {(1)}^{45 - 1} }{40 \times {(1)}^{40 - 1} } \: \: (by \: above \: formula)$

$= \displaystyle \: \frac{45 }{40 }$

$= \displaystyle \: \frac{9 }{8 }$

━━━━━━━━━━━━━━━━

Learn more from Brainly :-

1. Lthe value of limited x to 0 sin7x+sin5x÷tan5x-tan2x is

https://brainly.in/question/28582236

2. Lim x→0 (log 1+8x)/x

https://brainly.in/question/18998465