#lim_{x to infty}({2x-3}/{2x+5})^{2x+1}#
by the inverse property #e^{\ln x}=x#,
#=lim_{x to infty}e^{ln({2x-3}/{2x+5})^{2x+1}}#
by the log property #lnx^r=r lnx#,
#=lim_{x to infty}e^{(2x+1)ln({2x-3}/{2x+5})}#
by #2x-3={1}/{1/{2x-3}}#,
#=lim_{x to infty}e^{ln({2x-3}/{2x+5})/{1/{2x+1}}}#
by moving the limit inside the exponential function,
#=e^{lim_{x to infty}ln({2x-3}/{2x+5})/{1/{2x+1}}}#
by l'H#hat{"o"}#pital's Rule (0/0),
#=e^{lim_{x to infty}{{2x+5}/{2x-3}cdot{2cdot(2x+5)-(2x-3)cdot2}/{(2x+5)^2}}/{-2/(2x+1)^2}}#
by cleaning up a bit,
#=e^{lim_{x to infty}{-8(2x+1)^2}/{(2x-3)(2x+5)}}#
by dividing the numerator and the denominator by #x^2#,
#=e^{-8lim_{x to infty}{(2+1/x)^2}/{(2-3/x)(2+5/x)}}#
by #1/x to 0# as #x to infty#,
#=e^{-8cdot2^2/{2cdot 2}}=e^{-8}#
I hope that this was helpful.
