$\lim_{x\rightarrow 0^+}(x \int ^1_x \frac{\cos t }{t^2} \, dt)$ Use L'Hospital rule compute the following limit: 
$$\lim_{x\rightarrow 0^+} \left(x \int ^1_x \frac{\cos t }{t^2} \, dt\right)$$
my attempt:
$$\lim_{x\rightarrow 0^+} \frac{\displaystyle \left(\int ^1_x \frac{\cos t }{t^2} \, dt\right)} x$$ ($0/0$ form)
$$=\lim _{x\to0^+}\frac{\cos x/x^2}{-x^2}=-1$$
 A: It seems like you've almost got it! So you have
$$\lim_{x\rightarrow 0^+} \frac{\displaystyle \left(\int ^1_x \frac{\cos t }{t^2} \, dt\right)}{(1/x)}$$
Which is a good step. Then you apply L'Hopital's rule:
$$\lim_{x\rightarrow 0^+} \frac{\displaystyle \frac{d}{dx} \left(\int ^1_x \frac{\cos t }{t^2} \, dt\right)}{\displaystyle\frac{d}{dx} \left( \frac{1}{x}\right)}$$
Observe that $\frac{d}{dx} \left(\int ^1_x \frac{\cos t }{t^2} \, dt\right) = -\frac{d}{dx} \left(\int ^x_1 \frac{\cos t }{t^2} \, dt\right) = -\frac{\cos x }{x^2}$ and $\frac{d}{dx} \left(\frac{1}{x}\right) = -\frac{1}{x^2}$. This gives,
$$\lim _{x\to0^+}\frac{\cos x}{-x^2} \large/\frac{-1}{x^2} = \lim _{x\to0^+} \cos x = 1$$
So just a couple mistakes on your part. The first was writing $x$ instead of $1/x$ in the denominator, and the second when you applied the FTC to the integral which put you off by a minus sign.
A: Note that using integration by parts we have $$\int_{x} ^{1}\frac{\cos t} {t^{2}}\,dt=\frac{\cos x} {x} - \cos 1-\int_{x}^{1}\frac{\sin t} {t} \, dt$$ and multiplying by $x$ and taking limit as $x\to 0^{+}$ we can see that the limit is $1$.
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\mc}[1]{\mathcal{#1}}
 \newcommand{\mrm}[1]{\mathrm{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
\begin{align}
\lim_{x \to 0^{\large +}}\bracks{x\int_{x}^{1}{\cos\pars{t} \over t^{2}}\,\dd t} & =
\lim_{x \to 0^{\large +}}\bracks{%
x\int_{x}^{1}{\cos\pars{t} - 1 \over t^{2}}\,\dd t +
x\int_{x}^{1}{\dd t \over t^{2}}}
\\[5mm] & =
\lim_{x \to 0^{\large +}}\bracks{%
x\int_{x}^{1}{\cos\pars{t} - 1 \over t^{2}}\,\dd t - x + 1} =\bbx{1}
\end{align}
