Question

Find the following limits algebraically or explain why they don’t exist.

lim x->0 sin5x/2x
lim x->0 1-cosx/x
lim x->7 |x-7|/x-7
lim x->7 (/x+2)-3/x-7
lim h->0 (2+h)^3-8/h
lim t->0 1/t - 1/t^2+t

Answer 1

a lot of your questions contain ambiguous typing
e.g. the second last one probably says

lim ( (2+h)^3 - 8)/h as h --> 0

You could expand the top, or recognize it as a difference of cubes
recall A^3 - B^3 = (A-B)(A^2 + AB + B^2)

= lim [(2+h - 2)((2+h)^2 + 2(2+h) + 4) )/h , h--->0
= lim (2+h)^2 + 2(2+h) + 4 , h--> 0
= 4 + 4 + 4 = 12

for the first one
I will again assume you meant
lim sin (5x) / (2x) , as x --> 0

recall that lim sinØ/Ø = 1 as Ø -->0

so let's "construct" this pattern
multiply top and bottom by (5/2)
so
lim sin (5x) / (2x) , as x --> 0
= lim (5/2)sin (5x)/( (2x)(5/2)
= (5/2) lim sin (5x) / (5x) , as x--->0
= (5/2)(1) = 5/2

try some of the others now after checking on your typing using brackets.