<s>[INST] Evaluate: $\lim_{y\to x} \dfrac {y\sec y - x\sec x}{y-x}$ Evaluate: $\lim_{y\to x} \dfrac {y\sec y - x\sec x}{y-x}$
My Attempt:
$$=\lim_{y\to x} \dfrac {y\sec y - x\sec x}{y-x}$$
$$=\lim_{y\to x} \dfrac {y\cos x-x\cos y}{\cos x\cos y \cdot (y-x)}$$
 [/INST] $\lim_\limits{y\to x} \frac {y\sec y - x\sec x}{y-x}$ is the definition of $\frac {d}{dx} x\sec x$
But since you requested a solution to this limit that was algebraic, here goes: 
$\frac {y\cos x - x\cos y}{(y-x)\cos x\cos y}\\
\frac {y\cos x - y\cos y + y \cos y- x\cos y}{(y-x)\cos x\cos y}\\
\frac {y(\cos x - \cos y) + (y-x)\cos y}{(y-x)\cos x\cos y}\\
\frac {-2y \sin(\frac {y+x}{2})\sin (\frac {x-y}{2})} {(y-x)\cos x\cos y}+\frac {1}{\cos x}\\
\frac {y\sin(\frac {y+x}{2})}{\cos x\cos y}\frac {(-2 \sin (\frac {x-y}{2}))} {(y-x)}+\frac {1}{\cos x}\\
\lim_\limits{y\to x}\frac {y\sin(\frac {y+x}{2})}{\cos x\cos y}\lim_\limits{y\to x}\frac {(2 \sin (\frac {y-x}{2}))} {(y-x)} + \lim_\limits{y\to x}\frac {1}{\cos x} = \frac {x\sin x}{\cos^2 x} + \frac{1}{\cos x}\\
x\sec x\tan x + \sec x$ 
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