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7*tan(9*x/5)/x
Limit of the function/ 7*tan(9*x/5)/x

Limit of the function 7*tan(9*x/5)/x

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     /     /9*x\\
     |7*tan|---||
     |     \ 5 /|
 lim |----------|
x->0+\    x     /
lim⁡x→0+(7tan⁡(9x5)x)\lim_{x \to 0^+}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right) 
Limit((7*tan((9*x)/5))/x, x, 0)
Detail solution
Let's take the limit
lim⁡x→0+(7tan⁡(9x5)x)\lim_{x \to 0^+}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right)
transform
True

=
lim⁡x→0+(7tan⁡(9x5)x)lim⁡x→0+1cos⁡(9x5)=lim⁡x→0+(7tan⁡(9x5)x)\lim_{x \to 0^+}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right) \lim_{x \to 0^+} \frac{1}{\cos{\left(\frac{9 x}{5} \right)}} = \lim_{x \to 0^+}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right)
Do replacement
u=9x5u = \frac{9 x}{5}
then
lim⁡x→0+(7sin⁡(9x5)x)=lim⁡u→0+(63sin⁡(u)5u)\lim_{x \to 0^+}\left(\frac{7 \sin{\left(\frac{9 x}{5} \right)}}{x}\right) = \lim_{u \to 0^+}\left(\frac{63 \sin{\left(u \right)}}{5 u}\right)
=
63lim⁡u→0+(sin⁡(u)u)5\frac{63 \lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)}{5}
The limit
lim⁡u→0+(sin⁡(u)u)\lim_{u \to 0^+}\left(\frac{\sin{\left(u \right)}}{u}\right)
is first remarkable limit, is equal to 1.

The final answer:
lim⁡x→0+(7tan⁡(9x5)x)=635\lim_{x \to 0^+}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right) = \frac{63}{5}
Lopital's rule
We have indeterminateness of type
0/0,

i.e. limit for the numerator is
lim⁡x→0+tan⁡(9x5)=0\lim_{x \to 0^+} \tan{\left(\frac{9 x}{5} \right)} = 0
and limit for the denominator is
lim⁡x→0+(x7)=0\lim_{x \to 0^+}\left(\frac{x}{7}\right) = 0
Let's take derivatives of the numerator and denominator until we eliminate indeterninateness.
lim⁡x→0+(7tan⁡(9x5)x)\lim_{x \to 0^+}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right)
=
Let's transform the function under the limit a few
lim⁡x→0+(7tan⁡(9x5)x)\lim_{x \to 0^+}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right)
=
lim⁡x→0+(ddxtan⁡(9x5)ddxx7)\lim_{x \to 0^+}\left(\frac{\frac{d}{d x} \tan{\left(\frac{9 x}{5} \right)}}{\frac{d}{d x} \frac{x}{7}}\right)
=
lim⁡x→0+(63tan⁡2(9x5)5+635)\lim_{x \to 0^+}\left(\frac{63 \tan^{2}{\left(\frac{9 x}{5} \right)}}{5} + \frac{63}{5}\right)
=
lim⁡x→0+(63tan⁡2(9x5)5+635)\lim_{x \to 0^+}\left(\frac{63 \tan^{2}{\left(\frac{9 x}{5} \right)}}{5} + \frac{63}{5}\right)
=
635\frac{63}{5}
It can be seen that we have applied Lopital's rule (we have taken derivatives with respect to the numerator and denominator) 1 time(s)
The graph
02468-8-6-4-2-1010-250250
Plot the graph
One‐sided limits [src] 
     /     /9*x\\
     |7*tan|---||
     |     \ 5 /|
 lim |----------|
x->0+\    x     /
lim⁡x→0+(7tan⁡(9x5)x)\lim_{x \to 0^+}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right) 
63/5
635\frac{63}{5} 
= 12.6
     /     /9*x\\
     |7*tan|---||
     |     \ 5 /|
 lim |----------|
x->0-\    x     /
lim⁡x→0−(7tan⁡(9x5)x)\lim_{x \to 0^-}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right) 
63/5
635\frac{63}{5} 
= 12.6
= 12.6
Other limits x→0, -oo, +oo, 1
lim⁡x→0−(7tan⁡(9x5)x)=635\lim_{x \to 0^-}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right) = \frac{63}{5}
More at x→0 from the left
lim⁡x→0+(7tan⁡(9x5)x)=635\lim_{x \to 0^+}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right) = \frac{63}{5}
lim⁡x→∞(7tan⁡(9x5)x)\lim_{x \to \infty}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right)
More at x→oo
lim⁡x→1−(7tan⁡(9x5)x)=7tan⁡(95)\lim_{x \to 1^-}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right) = 7 \tan{\left(\frac{9}{5} \right)}
More at x→1 from the left
lim⁡x→1+(7tan⁡(9x5)x)=7tan⁡(95)\lim_{x \to 1^+}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right) = 7 \tan{\left(\frac{9}{5} \right)}
More at x→1 from the right
lim⁡x→−∞(7tan⁡(9x5)x)\lim_{x \to -\infty}\left(\frac{7 \tan{\left(\frac{9 x}{5} \right)}}{x}\right)
More at x→-oo
Rapid solution [src] 
63/5
635\frac{63}{5}
Expand and simplify
Numerical answer [src] 
12.6
12.6
The graph
Limit of the function 7*tan(9*x/5)/x
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