Question

Try Yourself 1) Evaluate the following limits (x + 1)4 - 24 lim x-1(2x + 1,5 - 35

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Answer 1

Step-by-step explanation:

We have,

$\lim_{x \rarr1} \frac{(x + 1) ^{4} - {2}^{4} }{(2x + 1)^{5} - 3^{5} }$

$= \lim_{x \rarr1} \frac{(x + 1) ^{4} - {2}^{4} }{(2x + 1)^{5} - 3^{5} }. \frac{x - 1}{x - 1}$

$= \lim_{x \rarr1} \frac{(x + 1) ^{4} - {2}^{4} }{x - 1 }. \frac{x - 1}{(2x + 1)^{5} - {3}^{5} }$

$= \lim_{x \rarr1} \frac{(x + 1) ^{4} - {2}^{4} }{(x + 1) - 2 }. \frac{1}{2} \lim_{x \rarr1} \frac{2x - 2}{(2x + 1)^{5} - {3}^{5} }$

$= \lim_{x \rarr1} \frac{(x + 1) ^{4} - {2}^{4} }{(x + 1) - 2 }. \frac{1}{2} \lim_{x \rarr1} \frac{(2x + 1) - 3}{(2x + 1)^{5} - {3}^{5} }$

$= \lim_{x \rarr1} \frac{(x + 1) ^{4} - {2}^{4} }{(x + 1) - 2 }. \frac{1}{2} \frac{1}{ \lim_{x \rarr1} \dfrac{(2x + 1)^{5} - {3}^{5}}{ (2x + 1) - 3} }$

$= (4 .{2}^{3} ). \frac{1}{2} . \frac{1}{5 .{3}^{4} }$

$= \frac{4 \times 8}{5 \times {3}^{4} \times 2 }$

$= \frac{16}{5 \times {3}^{4} }$

$= \frac{16}{5 \times 81 }$

$= \frac{16}{405 }$

Answer 2

Answer:16/405

Step-by-step explanation:

Using L'Hospital's rule:

Step 1: differentiate first.

Step 2: after differentiation put the value of the limit and here comes your answer.