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Exercise

$\lim_{x\to\infty\:}\left(\frac{4x+4}{4x+9}\right)^{-2x+3}$

Step-by-step Solution

1

Factor the polynomial $4x+4$ by it's greatest common factor (GCF): $4$

$\lim_{x\to\infty }\left(\left(\frac{4\left(x+1\right)}{4x+9}\right)^{\left(-2x+3\right)}\right)$
2

Rewrite the limit using the identity: $a^x=e^{x\ln\left(a\right)}$

$\lim_{x\to\infty }\left(e^{\left(-2x+3\right)\ln\left(\frac{4\left(x+1\right)}{4x+9}\right)}\right)$
3

Multiplying polynomials $\ln\left(\frac{4\left(x+1\right)}{4x+9}\right)$ and $-2x+3$

$\lim_{x\to\infty }\left(e^{\left(-2x\ln\left(\frac{4\left(x+1\right)}{4x+9}\right)+3\ln\left(\frac{4\left(x+1\right)}{4x+9}\right)\right)}\right)$
4

Simplify $e^{\left(-2x\ln\left(\frac{4\left(x+1\right)}{4x+9}\right)+3\ln\left(\frac{4\left(x+1\right)}{4x+9}\right)\right)}$ by applying the properties of exponents and logarithms

$\lim_{x\to\infty }\left(\left(\frac{4\left(x+1\right)}{4x+9}\right)^{-2x}e^{3\ln\left(\frac{4\left(x+1\right)}{4x+9}\right)}\right)$
5

Evaluate the limit $\lim_{x\to\infty }\left(\left(\frac{4\left(x+1\right)}{4x+9}\right)^{-2x}e^{3\ln\left(\frac{4\left(x+1\right)}{4x+9}\right)}\right)$ by replacing all occurrences of $x$ by $\infty $

0

Final answer to the exercise

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$-\left|-68\right|$
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