f[x_]:=ArcTan[x]

Plot[f[x],{x,-20,20}]

{Limit[f[x],x->Infinity],Limit[f[x],x->-Infinity]}

Clear[f];
f[x_]:= E^(-x/10)Cos[x^2];
Plot[f[x],{x,-12,24},PlotRange->All]

{Limit[f[x],x->Infinity],Limit[f[x],x->-Infinity]}

{0,Interval[{-∞,∞}]}

Clear[f];
f[x_]:= (2x^2+5x-1)/(x^2+1);
Plot[{f[x],2},{x,-20,20},PlotRange->All]
{Limit[f[x],x->Infinity],Limit[f[x],x->-Infinity]}

Clear[f];
f[x_]= Limit[(Sqrt[x+h]-Sqrt[x])/h,h->0]

Limit[Sum[Cos[k Pi/(3n)](Pi/(3n)),{k,1,n}],n->Infinity]

Limit[Sum[Cos[k x/n](x/n),{k,1,n}],n->Infinity]

Clear[f];
f[x_]:= -x(x-2);
secLine[a_,h_][x_]:=f[a]+(f[a+h]-f[a])/h(x-a)

q[x_]=secLine[0.5,0.5][x]

Plot[{f[x],q[x]},{x,0,2}]

Plot[{f[x],q[x],f[0.5]+f'[0.5](x-0.5)},{x,0,2}]

Plot[{f[x],q[x],f[0.75]+f'[0.75](x-0.75)},{x,0,2}]

Clear[f];
f[x_]=x^3;
f1[x_]=D[f[x],{x,1}];
plotTan[y_]:=Plot[{f[x],f[y]+f1[y](x-y)},{x,y-1,y+1}];

plotTan[2.5]

g[x_]=x Sin[x];
 Limit[(g[x+h]-g[x])/h,h->0]

g'[x]

D[g[x],x]

(∂_x)g[x]

(∂_x){c u[x],u[x]+v[x],u[x] v[x],u[x]/v[x]}//TableForm

({{c u'[x]}, {u'[x]+v'[x]}, {v[x] u'[x]+u[x] v'[x]}, {((u'[x])/(v[x]))-((u[x] v'[x])/((v[x])^2))}})

Clear[g];
g[x_]=x^2 Cos[a x];
g''[x]
D[g[x],{x,2}]
(∂_(x,x))g[x]

Table[D[g[x],{x,k}],{k,0,5}]//TableForm

Integrate[Cos[x],x]

∫Cos[x]\[DifferentialD]x

∫1/(x^2+3)\[DifferentialD]x

Integrate[Cos[x],{x,0,Pi/3}]