# Problems and Solutions of Basic Limits:

Here we will discuss various basic problems of limits with solutions.
Example 1: Evaluate lim_{x to 0} (sin x + cos x)
Solution:
lim_{x to 0} (sin x +cos x)
=lim_{x to 0} sin x + lim_{x to 0}cos x
=sin 0 + cos 0
=0+1
=1.
Example 2: Evaluate lim_{x to 1} sin(3x^2-2x-1)
Solution:
lim_{x to 1} sin(3x^2-2x-1)
=sin[lim_{x to 1}(3x^2-2x-1)]
=sin[lim_{x to 1}3x^2 -lim_{x to1} 2x – lim_{x to 1}1]
=sin[3 cdot 1^2-2 cdot 1-1]
=sin 0=0.
[Formula used: lim_{x to a}sin[f(x)]=sin[lim_{x to a} f(x)]]
Example 3: Evaluate lim_{x to 0} e^{2x^2-x+1}
Solution:
lim_{x to 0} e^{2x^2-x+1}
=e^{lim_{x to 0}(2x^2-x+1)}
=e^{[lim_{x to 0} 2x^2-lim_{x to 0}x+ lim_{x to 0}1]}
=e^{2 cdot 0^2-0+1}=e^1=e
[Formula used: lim_{x to a}e^{f(x)}=e^{lim_{x to a} f(x)}]
Example 4: Evaluate lim_{x to 1} frac{x^2-1}{x-1}
Solution:
lim_{x to 1} frac{x^2-1}{x-1}
=lim_{x to 1} frac{(x-1)(x+1)}{x-1}
=lim_{x to 1} (x+1)
=1+1=2
[Formula used: a^2-b^2=(a-b)(a+b)]
Example 5: Evaluate lim_{x to -2} frac{x^3+8}{x+2}
Solution:
lim_{x to -2} frac{x^3+8}{x+2}
=lim_{x to -2}frac{x^3-(-2)^3}{x-(-2)}
=3 cdot (-2)^{3-1}=12
[Formula used: lim_{x to a}frac{x^n-a^n}{x-a}=na^{n-1}
In the above example, a=-2 and n=3]