Question

1. evaluate the limits

a) \\( \lim _ { x \
ightarrow \infty } \frac { e ^ { x } + 5 x } { x ^ { 2 } + 5 } \\)
b) \\( \lim _ { x \
ightarrow 0 } \frac { \sin ( 3 x ) - 3 x } { x ^ { 3 } } \\)
c) \\( \lim _ { x \
ightarrow \infty } x \sin ( 2 / x ) \\)
d) \\( \lim _ { x \
ightarrow \infty } ( e ^ { 2 x } + 3 ) ^ { 3 / x } \\)
e) \\( \lim _ { x \
ightarrow \infty } \left( 1 + \frac { 3 } { x } \
ight) ^ { 2 x } \\)

Explanation:

Step1: Apply L'Hopital's Rule for a)

Since $\lim_{x
ightarrow\infty}\frac{e^{x}+5x}{x^{2}+5}$ is in the $\frac{\infty}{\infty}$ form.
Differentiate numerator and denominator: $\lim_{x
ightarrow\infty}\frac{e^{x}+5}{2x}$.
Still in $\frac{\infty}{\infty}$ form.
Differentiate again: $\lim_{x
ightarrow\infty}\frac{e^{x}}{2}=\infty$.

Step2: Use Taylor series for b)

The Taylor series of $\sin t=t-\frac{t^{3}}{6}+\frac{t^{5}}{120}-\cdots$. Let $t = 3x$, then $\sin(3x)=3x-\frac{(3x)^{3}}{6}+\frac{(3x)^{5}}{120}-\cdots$.
$\lim_{x
ightarrow0}\frac{\sin(3x)-3x}{x^{3}}=\lim_{x
ightarrow0}\frac{3x-\frac{27x^{3}}{6}+\cdots - 3x}{x^{3}}=\lim_{x
ightarrow0}\frac{-\frac{9x^{3}}{2}+\cdots}{x^{3}}=-\frac{9}{2}$.

Step3: Use substitution for c)

Let $t=\frac{1}{x}$, as $x
ightarrow\infty$, $t
ightarrow0$. Then $\lim_{x
ightarrow\infty}x\sin(\frac{2}{x})=\lim_{t
ightarrow0}\frac{\sin(2t)}{t}$.
Using $\lim_{u
ightarrow0}\frac{\sin u}{u} = 1$, with $u = 2t$, we get $\lim_{t
ightarrow0}\frac{\sin(2t)}{t}=2\lim_{t
ightarrow0}\frac{\sin(2t)}{2t}=2$.

Step4: Take natural logarithm for d)

Let $y=(e^{2x}+3)^{\frac{3}{x}}$, then $\ln y=\frac{3}{x}\ln(e^{2x}+3)$.
$\lim_{x
ightarrow\infty}\ln y=\lim_{x
ightarrow\infty}\frac{3\ln(e^{2x}+3)}{x}$.
Apply L'Hopital's Rule: $\lim_{x
ightarrow\infty}\frac{3\frac{2e^{2x}}{e^{2x}+3}}{1}=6$.
Since $y = e^{\ln y}$, $\lim_{x
ightarrow\infty}y=e^{6}$.

Step5: Use the formula $\lim_{x

ightarrow\infty}(1 + \frac{a}{x})^{bx}=e^{ab}$ for e)
Here $a = 3$ and $b = 2$, so $\lim_{x
ightarrow\infty}(1+\frac{3}{x})^{2x}=e^{6}$.

Answer:

a) $\infty$
b) $-\frac{9}{2}$
c) $2$
d) $e^{6}$
e) $e^{6}$