Vrftsjtryk

I am having a little trouble going about:
$$lim_{xto infty} left(frac{14x}{14x+10}right)^{10x}$$

Using $ln$ properties we can bring down the $10x$ exponent and have:$$ln y=10xlnleft(frac{14x}{14x+10}right)$$

And from here I get stuck trying to apply L'Hospital's Rule to find the limit.

Hint :

$$10x ln left(frac{14x}{14x+10}right)= frac{ln left(frac{14x}{14x+10}right)}{frac{1}{10x}}$$

Hint: You want to reach some kind of indeterminate form, so you need to mess around with the fractions to reach a desirable form.

$$ln y=10xlnleft(frac{14x}{14x+10}right)$$

$$ln y=frac{lnleft(frac{14x}{14x+10}right)}{frac{1}{10x}}$$

Now, you can continue.

As an alternative approach (both faster and easier), you can use $lim_limits{n to infty}big(1+frac{x}{n}big)^n = e^x$. These expressions are easily manipulated to reach such a form.

Notice the numerator is $10$ less than the denominator, so $frac{14x}{14x+14} = 1+frac{-10}{14x+10}$. Hence, you get

$$lim_{xto infty} left(frac{14x}{14x+10}right)^{10x} = lim_{xto infty} Biggl[left(1+frac{-10}{14x+10}right)^{14x+10}Biggl]^{frac{10x}{14x+10}}$$

HINT

We can use that

$$left(frac{14x}{14x+10}right)^{10x}=left(frac{14x+10-10}{14x+10}right)^{10x}=left[left(1-frac{10}{14x+10}right)^{frac{14x+10}{10}}right]^{frac{10cdot 10x}{14x+10}}$$

and refer to standard limits.

Once you get to the logarithm, you need
$$
lim_{xtoinfty}frac{lndfrac{14x}{14x+10}}{dfrac{1}{10x}}
$$
which is in the form $0/0$ and l'Hôpital can be applied. However it's much simpler if the numerator is written as $ln14+ln x-ln(14x+10)$, in order to compute the derivative (you know the limit is $0$ anyhow). So we get
$$
lim_{xtoinfty}frac{dfrac{1}{x}-dfrac{14}{14x+10}}{-dfrac{1}{10x^2}}=
lim_{xtoinfty}-10x^2frac{14x+10-14x}{x(14x+10)}=lim_{xtoinfty}-frac{100x}{14x+10}
$$
When you have this limit, let me call it $l$, the one you started with is $e^l$.

Tips:

With equivalents, it 's very fast:$$lnbiggl(frac{14x}{14x+10}biggr)=lnbiggl(1-frac{10}{14x+10}biggr)sim_infty -frac{10}{14x+10}sim_infty -frac{10}{14x}=-frac 57,$$
so,as equivalence is compatible with multplication/division,
$$10xlnbiggl(frac{14x}{14x+10}biggr)sim_infty -frac{50x}{7x}to -frac{50}7.$$
Warning: equivalence is not compatible with addition/subtraction. It is compatible, under mild hypotheses, with composition on the left by logarithm.