Check my answer

The first one is, perhaps, more complicated than necessary:

$$\lim_{x\rightarrow 0}(\cot x - \frac{1}{x})$$

$$\lim_{x \rightarrow 0}(\frac{\cos x}{\sin x}-\frac{1}{x})$$

As x → 0, cos x → 1 and sin x → x, so we get:

$$\lim_{x \rightarrow 0}(\frac{1}{x} - \frac{1}{x})=\lim_{x \rightarrow 0}(0) = 0$$

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The second one is ok.

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Also this 

thank you alan 

What about this how can I solve it 

And for the second picture I found that when x goes to infinity fx goes to infinity ! 

but the graph doesn't show that !!! 

https://www.desmos.com/calculator/f49yopuqer

$$\lim_{x \rightarrow 0}(\frac{x+1}{x}-\frac{2}{sin(2x)})$$

Write this as:

$$\lim_{x \rightarrow 0}(1+\frac{1}{x}-\frac{2}{sin(2x)})$$

as x tends to 0, sin(2x) tends to 2x, so 2/2x tends to 1/x:

$$\lim_{x \rightarrow 0}(1+\frac{1}{x}-\frac{1}{x})=1$$

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The graph of e^x/x⁴ does go to infinity as x goes to infinity.  It does so so rapidly, that the values get too large to show for the range of x's you can see!

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thank you