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`]